0411116v3 [math.DG] 26 Oct 2009

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Oct 26, 2009 - Suppose that a coassociative 4-fold N in R7 is asymptotically conical to a ...... c , corresponding to 4s2 − 5r2 > 0 and 4s2 − 5r2 < 0 respectively.
arXiv:math/0411116v3 [math.DG] 26 Oct 2009

Deformation Theory of Asymptotically Conical Coassociative 4-folds Jason D. Lotay University College Oxford Abstract Suppose that a coassociative 4-fold N in R7 is asymptotically conical to a cone C with rate λ < 1. If λ ∈ [−2, 1) is generic, we show that the moduli space of coassociative deformations of N which are also asymptotically conical to C with rate λ is a smooth manifold, and we calculate its dimension. If λ < −2 and generic, we show that the moduli space is locally homeomorphic to the kernel of a smooth map between smooth manifolds, and we give a lower bound for its expected dimension. We also derive a test for when N will be planar if λ < −2 and we discuss examples of asymptotically conical coassociative 4-folds.

1

Introduction

In this article we study coassociative 4-folds N in R7 which are asymptotically conical (AC). Our main result (cf. Theorem 6.4 & Corollary 6.5) is the following. Theorem 1.1 Let N be a coassociative 4-fold in R7 which is AC with rate λ < 1 to a cone C. Let M(N, λ) be the moduli space of coassociative deformations of N which are also AC with rate λ to C. (a) For generic λ ∈ [−2, 1) the deformation theory is unobstructed so M(N, λ) is a smooth manifold near N . (b) For generic λ < −2, M(N, λ) is locally homeomorphic to the kernel of a smooth map between smooth manifolds. 2000 Mathematics Subject Classification: 53C38.

1

In case (a), the dimension of M(N, λ) is equal to the dimension of the infinitesimal deformation space. We determine this dimension explicitly in Proposition 7.12 if additionally λ ∈ [−2, 0), and we give upper and lower bounds for it otherwise in Proposition 7.13. In case (b), the map can be considered as a projection from the infinitesimal deformation space to the obstruction space and hence M(N, λ) is smooth if the latter space is zero. Moreover, we give a lower bound for the expected dimension of M(N, λ) for λ < −2 in Proposition 7.14. These dimensions are given in terms of topological data from N and C, and an analytic quantity determined by the cone C. This article is motivated by the work of McLean [23, §4] on compact coassociative 4-folds and Marshall [21] on deformations of AC special Lagrangian (SL) submanifolds. The latter was in fact studied earlier by Pacini [24] using different methods. Asymptotically conical SL submanifolds are also discussed in the papers by Joyce [5]-[9] on SL submanifolds with conical singularities. Deformations of asymptotically cylindrical coassociative 4-folds are studied by Joyce and Salur in [11] and Salur in [25]. Other coassociative deformation theories have been studied by the author: coassociative 4-folds with conical singularities in [18] and compact coassociative 4-folds with boundary in [12] (with Kovalev). We begin, in §2, by defining the submanifolds that we study here. Much of our work is analytic in nature, so to obtain many of the results we use weighted Sobolev spaces, which are a natural choice when studying AC submanifolds in this way. Thus, in §3, we define the weighted Banach spaces that we require. In §4 we construct a deformation map which corresponds locally to the moduli space M(N, λ). We may therefore view the kernel of the linearisation of the deformation map at zero as the infinitesimal deformation space. We also define an associated map which is elliptic at zero since its derivative there acts as d + d∗ from pairs of self-dual 2-forms and 4-forms to 3-forms. This allows to prove that the kernel of the deformation map consists of smooth forms. In §5 we discuss the Fredolm and index theory of the operator d + d∗ : in particular, we describe a countable discrete set D, depending only on C, consisting of the rates λ for which d + d∗ is not Fredholm. Our main result of the section (Theorem 5.10) identifies the obstruction space for our deformation theory. In §6 we prove the deformation theory results which lead directly to Theorem 1.1. Section §7 then contains our aforementioned dimension calculations. In §8, we construct two invariants of N and hence derive a test for when N will be planar. Finally, in §9, we discuss examples of AC coassociative 4-folds, including explicit examples which have rate −3/2. There are no known concrete examples of AC coassociative 4-folds with rate λ < −2, but such submanifolds 2

are essential for the desingularization theory of coassociative 4-folds with conical singularities, as discussed in [19]. Acknowledgements Many thanks are due to Dominic Joyce for his hard work and guidance during the course of this project. I would also like to thank EPSRC for providing the funding for this study.

2

Asymptotically conical coassociative submanifolds of R7

We begin by defining coassociative 4-folds in R7 , for which we introduce a distinguished 3-form on R7 , following the notation in [10, Definition 11.1.1]. Definition 2.1 Let (x1 , . . . , x7 ) be coordinates on R7 and write dxij...k for the form dxi ∧ dxj ∧ . . . ∧ dxk . Define a 3-form ϕ by: ϕ = dx123 + dx145 + dx167 + dx246 − dx257 − dx347 − dx356 . The 4-form ∗ϕ, where ϕ and ∗ϕ are related by the Hodge star, is given by: ∗ϕ = dx4567 + dx2367 + dx2345 + dx1357 − dx1346 − dx1256 − dx1247 . The subgroup of GL(7, R) preserving the 3-form ϕ is G2 . We can now characterise coassociative 4-folds in R7 . Definition 2.2 A 4-dimensional submanifold N of R7 is coassociative if and only if ϕ|N ≡ 0 and ∗ϕ|N > 0. This is not the standard definition, which is formulated in terms of calibrated geometry, but is equivalent to it by [3, Proposition IV.4.5 & Theorem IV.4.6]. Remark The vanishing of ϕ on a 4-fold N in R7 forces ∗ϕ to be nowhere vanishing on N . Thus, the condition ∗ϕ|N > 0 amounts to a choice of orientation. In order that we may study deformations of coassociative 4-folds, we need an important elementary result [23, Proposition 4.2]. Proposition 2.3 Let N be a coassociative 4-fold in R7 . There is an isomorphism N between the normal bundle ν(N ) of N in R7 and Λ2+ T ∗ N given by N : v 7→ (v · ϕ)|T N . Note Let u ∈ T R7 . There exist unique normal and tangent vectors v and w on N such that u|N = v + w. Since ϕ vanishes on the coassociative 4-fold N , (w · ϕ)|T N = 0. Thus, N (u) = N (v) ∈ Λ2+ T ∗ N . So, N : T R7 |N → Λ2+ T ∗ N . 3

Before defining AC submanifolds we clarify what we mean by a cone in Rn . Definition 2.4 A cone C ⊆ Rn is a nonsingular submanifold, except perhaps at 0, satisfying et C = C for all t ∈ R. We call Σ = C ∩ S n−1 the link of C. Definition 2.5 Let C be a closed cone in Rn , let Σ be the link of C and let N be a closed submanifold of Rn . Then N is asymptotically conical (AC) to C (with rate λ) if there exist constants λ < 1 and R > 1, a compact subset K of N , and a diffeomorphism Ψ : (R, ∞) × Σ → N \ K such that j   ∇ Ψ(r, σ) − ι(r, σ) = O rλ−j

for j ∈ N as r → ∞,

(1)

where ι : (0, ∞) × Σ → Rn is the inclusion map given by ι(r, σ) = rσ. Here | . | is calculated using the conical metric gcone = dr2 + r2 gΣ on (0, ∞) × Σ, where gΣ is the round metric on S n−1 restricted to Σ, and ∇ is a combination of the Levi–Civita connection derived from gcone and the flat connection on Rn , which acts as partial differentiation. We also make the following definition. Definition 2.6 Let (M, g) be a Riemannian n-manifold. Then M is asymptotically conical (AC) (with rate λ) if there exist constants λ < 1 and R > 1, a compact (n−1)-dimensional Riemannian submanifold (Σ, gΣ ) of S n−1 , compact K ⊆ M , and a diffeomorphism Ψ : (R, ∞) × Σ → M \ K such that j ∗   ∇ Ψ (g) − g cone = O rλ−1−j

for j ∈ N as r → ∞,

(2)

where (r, σ) are coordinates on (0, ∞) × Σ, g cone = dr2 + r2 gΣ on (0, ∞) × Σ, ∇ is the Levi–Civita connection of g cone and | . | is calculated using gcone . Define ι : (0, ∞) × Σ → Rn by ι(r, σ) = rσ and let C = Im ι. Then C is the asymptotic cone of M and the components of M∞ = M \ K are the ends of M . The condition λ < 1 in the definition above ensures that, by (2), the metric g on M converges to gcone at infinity. Note By comparing (1) and (2), we see that if N is a Riemannian submanifold of Rn which is AC to a cone C with rate λ, it can be considered as an AC manifold with rate λ and asymptotic cone C. Definition 2.7 Let M be an AC manifold and use the notation of Definition 2.6. A radius function ρ : M → [1, ∞) on M is a smooth function such that there exist positive constants c1 < 1 and c2 > 1 with c1 r < Ψ∗ (ρ) < c2 r. 4

 If M is AC we may define a radius function ρ by setting ρ = 1 on K, ρ Ψ(r, σ) = r for r > R + 1, and then extending ρ smoothly to our required function on M . We conclude the section with the following elementary result. Proposition 2.8 Suppose that N is a coassociative 4-fold in R7 which is AC with rate λ to a cone C in R7 . Then C is coassociative. Proof: By Definition 2.2, we have that ϕ|N ≡ 0. We also have, using (1), that ∗   Ψ ϕ|N − ϕ|C = O rλ−1

as r → ∞.

Therefore, since λ − 1 < 0, ϕ|C → 0 as r → ∞. However, independent of r since Trσ C = Tσ C for all r > 0, σ ∈ Σ. Hence

ϕ|C must be ϕ|C ≡ 0. 

Notes (a) Manifolds are taken to be nonsingular and submanifolds to be embedded, for convenience, unless stated otherwise. (b) We use the convention that the natural numbers N = {0, 1, 2, . . .}.

3

Weighted Banach spaces

We shall define weighted Banach spaces of forms on an AC manifold, following [1, §1]. We use the notation and definition of the usual ‘unweighted’ Banach spaces as in [10, §1.2]; that is, Sobolev and H¨ older spaces are denoted by Lpk and C k, a respectively, where p ≥ 1, k ∈ N and a ∈ (0, 1). We also introduce the k for the space of forms ξ such that f ξ lies in C k for every smooth notation Cloc k, a compactly supported function f , and similarly define spaces Lpk, loc and Cloc . For the whole of this section, we let (M, g) be an AC n-manifold and ρ be a radius function on M as in Definition 2.7. Definition 3.1 Let p ≥ 1, k ∈ N and µ ∈ R. The weighted Sobolev space Lpk, µ (Λm T ∗ M ) of m-forms on M is the subspace of Lpk, loc (Λm T ∗ M ) such that

kξkLpk, µ

 k Z X  = j=0

M

 p1

|ρj−µ ∇j ξ|p ρ−n dVg 

is finite. The normed vector space Lpk, µ (Λm T ∗ M ) is a Banach space.

5

Note Lp (Λm T ∗ M ) = Lp0, − n (Λm T ∗ M ). p

(3)

We now define dual weighted Sobolev space which shall be invaluable later. Definition 3.2 Use the notation from Definition 3.1. Let p, q > 1 such that p 1 1 m ∗ p + q = 1, let k, l ∈ N and let µ ∈ R. Define a pairing h . , . i : Lk, µ (Λ T M ) × q m ∗ Ll, −n−µ (Λ T M ) → R by hξ, ηi = kξ ∧ ∗ηkL1 . We shall occasionally refer to this as the dual pairing. For our purposes, we take the dual space of Lpk, µ (Λm T ∗ M ) to be Lql, −n−µ (Λm T ∗ M ), with linear functionals represented by dual pairings. Definition 3.3 Let µ ∈ R and let k ∈ N. The weighted C k -space Cµk (Λm T ∗ M ) k of m-forms on M is the subspace of Cloc (Λm T ∗ M ) such that the norm kξkCµk =

k X

sup |ρj−µ ∇j ξ|

j=0 M

is finite. We also define Cµ∞ (Λm T ∗ M ) = ∩k≥0 Cµk (Λm T ∗ M ). Then Cµk (Λm T ∗ M ) is a Banach space but in general Cµ∞ (Λm T ∗ M ) is not. Notice that we have a continuous embedding Cµk ֒→ Cνl whenever k ≥ l and µ ≤ ν. We conclude this spate of definitions by defining weighted H¨ older spaces. These shall not be required for the majority of the paper as weighted Sobolev and C k -spaces will often suffice. Definition 3.4 Let a ∈ (0, 1), k ∈ N and µ ∈ R. Let d(x, y) be the geodesic distance between points x, y ∈ M , let 0 < c1 < 1 < c2 be constant and let H = {(x, y) ∈ M × M : x 6= y, c1 ρ(x) ≤ ρ(y) ≤ c2 ρ(x) and there exists a geodesic in M of length d(x, y) from x to y}, A section s of a vector bundle V on M , endowed with Euclidean metrics on its fibres and a connection preserving these metrics, is H¨ older continuous (with exponent a) if |s(x) − s(y)|V [s]a = sup < ∞. d(x, y)a (x,y)∈H We understand the quantity |s(x) − s(y)|V as follows. Given (x, y) ∈ H, there exists a geodesic γ of length d(x, y) connecting x and y. Parallel translation 6

along γ using the connection on V identifies the fibres over x and y and the metrics on them. Thus, with this identification, |s(x) − s(y)|V is well-defined. The weighted H¨ older space Cµk, a (Λm T ∗ M ) of m-forms ξ on M is the subspace k, a of Cloc (Λm T ∗ M ) such that the norm a kξkCµk, a = kξkCµk + [ξ]k, µ

is finite, where a [ξ]k, = [ρk+a−µ ∇k ξ]a . µ

Then Cµk, a (Λm T ∗ M ) is a Banach space. It is clear that we have an embedding Cµk, a (Λm T ∗ M ) ֒→ Cµl (Λm T ∗ M ) whenever l ≤ k. a The set H in the definition above is introduced so that [ξ]k, is well-defined. µ We shall need the analogue of the Sobolev Embedding Theorem for weighted spaces, which is adapted from [16, Lemma 7.2] and [1, Theorem 1.2].

Theorem 3.5 (Weighted Sobolev Embedding Theorem) Let p, q > 1, a ∈ (0, 1), µ, ν ∈ R and k, l ∈ N. (a) If k ≥ l, k − np ≥ l − nq and either p ≤ q and µ ≤ ν, or p > q and µ < ν, there is a continuous embedding Lpk, µ (Λm T ∗ M ) ֒→ Lql, ν (Λm T ∗ M ). (b) If k − np ≥ l + a, there is a continuous embedding Lpk, µ (Λm T ∗ M ) ֒→ Cµl, a (Λm T ∗ M ). We shall also require an Implicit Function Theorem for Banach spaces, which follows immediately from [14, Theorem 2.1]. Theorem 3.6 (Implicit Function Theorem) Let X and Y be Banach spaces and let U ⊆ X be an open neighbourhood of 0. Let F : U → Y be a C k -map (k ≥ 1) such that F (0) = 0. Suppose further that dF |0 : X → Y is surjective with kernel K such that X = K ⊕ A for some closed subspace A of X. There exist open sets V ⊆ K and W ⊆ A, both containing 0, with V × W ⊆ U , and a unique C k -map G : V → W such that F −1 (0) ∩ (V × W ) = in X = K ⊕ A.

4



 x, G(x) : x ∈ V

The deformation map

For the rest of the paper, let N ⊆ R7 be a coassociative 4-fold which is asymptotically conical to a cone C ⊆ R7 with rate λ, and use the notation of Definition 7

ι

2.5. In particular, (0, ∞) × Σ ∼ = C, where Σ = C ∩ S 6 , with coordinates (r, σ) Ψ ∼ N \ K, where K is a compact subset of N and on (0, ∞) × Σ, and (R, ∞) × Σ =

R > 1. Moreover, let ρ be a radius function on N , as defined in Definition 2.7, and choose Ψ uniquely by imposing the condition that Ψ(r, σ) − ι(r, σ) ∈ (Trσ C)⊥

for all (r, σ) ∈ (R, ∞) × Σ,

(4)

which can be achieved by making R and K larger if necessary. We shall endeavour to remind the reader of this notation when required.

4.1

Preliminaries

We wish to discuss deformations of N ; that is, coassociative submanifolds of R7 which are ‘near’ to N . We define this formally. Definition 4.1 The moduli space of deformations M(N, λ) is the set of coassociative 4-folds N ′ ⊆ R7 which are AC to C with rate λ such that there exists a diffeomorphism h : R7 → R7 , with h(N ) = N ′ , isotopic to the identity. The first result we need is immediate from the proof of [13, Chapter IV, Theorem 9]. Theorem 4.2 Let P be a closed submanifold of a Riemannian manifold M . There exist an open subset V of the normal bundle ν(P ) of P in M , containing the zero section, and an open set S in M containing P , such that the exponential map exp |V : V → S is a diffeomorphism. Note The proof of this result relies entirely on the observation that exp |ν(P ) is a local isomorphism upon the zero section. This information helps us prove a useful corollary.  Corollary 4.3 Let P = ι (R, ∞) × Σ , Q = N \ K and define nP : ν(P ) → R7 by nP (rσ, v) = v + Ψ(r, σ). There exist an open subset V of ν(P ), containing the zero section, and an open set S in R7 containing Q, such that nP |V : V → S is a diffeomorphism. Moreover, V can be chosen to be an open neighbourhood of the zero section in C11 . Proof: Note that nP takes the zero section of ν(P ) to Q. By the definition of Ψ in Definition 2.5, nP is a local isomorphism upon the zero section. Thus, by the note after Theorem 4.2, we have open sets V and S such that nP |V : V → S is a diffeomorphism.

8

Since Ψ − ι is orthogonal to (R, ∞) × Σ by (4), it can be identified with a small section of the normal bundle. Hence P lies in S as long as S grows with order O(r) as r → ∞. As we can form S and V in a translation equivariant way because we are working on a portion of the cone C, we can construct our sets with this growth rate at infinity and such that they do not collapse near R. Thus, we can ensure that V is an open set in C11 .  Recall that, by Proposition 2.8, C is coassociative. Therefore, since Ψ(r, σ)− ∼ νrσ (C) for r > R by (4), Ψ − ι can be identified with ι(r, σ) lies in (Trσ C)⊥ = the graph of an element γC of Λ2+ T ∗ P by Proposition 2.3, using the notation of Corollary 4.3. Then  for j ∈ N as r → ∞, (5) |∇jC γC | = O rλ−j

since N is AC to C with rate λ, where ∇C and | . | are the Levi–Civita connection and modulus calculated using the conical metric. Thus, γC lies in Cλ∞ (Λ2+ T ∗ P ). Moreover, we have a decomposition: R7 = TΨ(r,σ) N ⊕ νrσ (C)

at Ψ(r, σ). We can therefore identify νΨ(r,σ) (N ) with νrσ (C) and hence identify Λ2+ T ∗ N and Λ2+ T ∗ C near infinity. Formally, we have the following. Proposition 4.4 Use the notation of Corollary 4.3 and let C and N be the isomorphisms given by Proposition 2.3 applied to C and N respectively. There exists a diffeomorphism Υ : ν(P ) → ν(Q), with Υ(0) = 0, and hence a diffeo˜ : Λ2 T ∗ P → Λ2 T ∗ Q given by Υ ˜ = N ◦ Υ ◦ −1 . morphism Υ + + C Proposition 4.5 Use the notation of Corollary 4.3 and Proposition 4.4. There exist an open set U ⊆ Λ2+ T ∗ N containing the zero section and W = (N ◦Υ)(V ), a tubular neighbourhood T of N in R7 containing S, and a diffeomorphism δ : U → T , affine on the fibres, that takes the zero section of Λ2+ T ∗ N to N and such that the following diagram commutes: W

˜ −1 Υ /

−1 C

δ

 So

C (V )

nP

(6)

 V.

Moreover, we may choose U to be a C11 -open neighbourhood of the zero section. Proof: Define the diffeomorphism δ|W : W → S by (6). Interpolating smoothly over the compact set K ⊆ N , we extend S to T , W to U and δ|W to δ as required. Furthermore, by Corollary 4.3, V can be chosen to be an open neighbourhood of the zero section in C11 . Hence, we can arrange the same for U .  9

4.2

The map F and the associated map G

We introduce the notation Cλk (U ) = {α ∈ Cλk (Λ2+ T ∗ N ) : Γα ⊆ U }, where U is given by Proposition 4.5 and Γα is the graph of α. The fact that U is a C11 -open set ensures that Cλk (U ) is an open subset of Cλk (Λ2+ T ∗ N ) for k ≥ 1, since λ < 1. We use similar conventions to define subsets of the spaces discussed in §3, though we must take care the spaces contain continuous forms so that their graphs are well-defined. Moreover, these subsets will be open whenever the space embeds continuously in C11 ; for example, Lpk+1, λ (U ) is an open neighbourhood of zero in Lpk+1, λ (Λ2+ T ∗ N ) if p > 4 and k ≥ 1 by Theorem 3.5. We may now describe our deformation map. Definition 4.6 Use the notation from the start of this section and Proposition 1 (U ) let πα : N → Γα ⊆ U , where Γα is the graph of α, be given 4.5. For α ∈ Cloc  by πα (x) = x, α(x) . Let fα = δ ◦ πα and let Nα = fα (N ), so that Nα is the 1 0 deformation of N corresponding to α. We define F : Cloc (U ) → Cloc (Λ3 T ∗ N ) by  F (α) = fα∗ ϕ|Nα .

1 By Definition 2.2, Ker F is the set of α ∈ Cloc (U ) such that the deformation Nα of N is coassociative. Note that F is a nonlinear operator and that

dF |0 (α) = dα, 1 for all α ∈ Cloc (Λ2+ T ∗ N ), by [23, p. 731] and our choice of δ.

However, we only wish to consider smooth coassociative deformations Nα of N which are asymptotically conical to C with rate λ. If Nα is such a deformation, then α ∈ C ∞ (U ) and there exists a diffeomorphism Ψα : (R, ∞)×Σ → Nα \Kα , where Kα is a compact subset of Nα , as in Definition 2.5. We may define Ψα such that Ψα (r, σ) − ι(r, σ) is orthogonal to Trσ C for all σ ∈ Σ and r > R. Recall the notation of Corollary 4.3 and Proposition 4.4. Before Proposition 4.4, we showed that Ψ − ι can be identified with the graph of γC ∈ Cλ∞ (Λ2+ T ∗ P ) ˜ C ) ∈ C ∞ (Λ2 T ∗ Q). Similarly, Ψα − ι and therefore with the graph of αC = Υ(γ + λ can be identified with the graph of α + αC ∈ Cλ∞ (Λ2+ T ∗ Q). Hence, α ∈ Cλ∞ (U ). We conclude that Nα is AC to C with rate λ if and only if α ∈ Cλ∞ (U ). This leads immediately to our next result, which gives a local description of the moduli space M(N, λ) using the deformation map F . Proposition 4.7 Using the notation of Definitions 4.1 and 4.6, M(N, λ) is locally homeomorphic to the kernel of F : Cλ∞ (U ) → C ∞ (Λ3 T ∗ N ).

10

We now prove that we can rewrite F (α) as a sum of dα and a term which is no worse than quadratic in α and ∇α. This result will be useful throughout the article, but in particular we shall need it to derive regularity results in §4.3. Proposition 4.8 Use the notation of Definitions 3.1-3.4 and 4.6. We can write  (7) F (α)(x) = dα(x) + PF x, α(x), ∇α(x) for x ∈ N , where

PF : {(x, y, z) : (x, y) ∈ U, z ∈ Tx∗ N ⊗ Λ2+ Tx∗ N } → Λ3 T ∗ N  is a smooth map such that PF (x, y, z) ∈ Λ3 Tx∗ N . Denote PF x, α(x), ∇α(x) by PF (α)(x) for all x ∈ N . Let µ < 1. For each n ∈ N, if α ∈ Cµn+1 (U ) and kαkC11 is sufficiently small, n PF (α) ∈ C2µ−2 (Λ3 T ∗ N ) and there exists a constant cn > 0 such that n kPF (α)kC2µ−2 ≤ cn kαk2C n+1 .

(8)

µ

Let p > 4, k ≥ 1, l ∈ N and a ∈ (0, 1). If α ∈ Lpk+1, µ (U ) or α ∈ Cµl+1, a (U ),

l, a with kαkC11 sufficiently small, PF (α) ∈ Lpk, 2µ−2 (Λ3 T ∗ N ) or C2µ−2 (Λ3 T ∗ N ) and there exist constants lp, k > 0 and cl, a > 0 such that

kPF (α)kLpk, 2µ−2 ≤ lp, k kαk2Lp

k+1, µ

or

kPF (α)kC l, a ≤ cl, a kαk2C l+1, a . 2µ−2

n C2µ−2

µ

n Cµ−1 .

Remarks As µ < 1, 2µ − 2 < µ − 1, so ֒→ Similar continuous embeddings occur for the weighted Sobolev and H¨ older spaces. Furthermore, the conditions p > 4, k ≥ 1 and l ∈ N ensure, by Definition 3.4 and Theorem 3.5(b), that Cµl+1, a ֒→ C11 and Lpk+1, µ ֒→ C11 . Proof: First, by the definition of F , F (α)(x) relates to the tangent space to the  graph Γα of α at πα (x) = x, α(x) . Note that Tπα (x) Γα depends on both α(x) and ∇α(x) and hence so must F (α)(x). We may then define PF by (7) such that it is a smooth function of its arguments as claimed. We only prove the esimate (8) on PF as the results for weighted Sobolev and H¨ older spaces can be deduced from the work presented here. Recall the notation at the start of this section and of Corollary 4.3 and Proposition 4.4. Let ∇C denote the Levi–Civita connection of the conical metric on C. We argued after Corollary 4.3 that we may identify the displacement Ψ − ι of N from C, outside 1 (U ), there the compact subset K, with γC ∈ Cµ∞ (Λ2+ T ∗ P ). For each α ∈ Cloc 1 2 ∗ ˜ exists a unique γ ∈ Cloc (Λ+ T P ) such that α = Υ(γ) on N \ K. Thus, define 1 a function FC (γ + γC ), for γ ∈ Cloc (Λ2+ T ∗ P ), on (R, ∞) × Σ by  FC (γ + γC )(r, σ) = F (α) Ψ(r, σ) , (9) 11

˜ where α|N \K = Υ(γ). Now define a smooth function PC by an equation analogous to (7): FC (γ + γC )(r, σ)  = d(γ + γC )(r, σ) + PC (r, σ), (γ + γC )(r, σ), ∇C (γ + γC )(r, σ) .

(10)

We notice that FC and PC are only dependent on the cone C and, rather trivially, on R. Therefore, because of this fact and our choice of δ in Proposition 4.5, these functions have scale equivariance properties. We may therefore derive equations and inequalities on {R} × Σ and deduce the result on all of (R, ∞)× Σ by introducing an appropriate scaling factor of r. Now, since the graph of α = 0 corresponds to our coassociative 4-fold N ,  ˜ F Υ(0) = F (0) = 0. So, taking γ = 0 in (9) gives: FC (γC ) = dγC + PC (γC ) = 0,

(11)

adopting similar notation for PC (γC ) as for PF (α). From (7)-(11) we calculate:  PF (α) Ψ(r, σ) = dγC (r, σ) + PC (γ + γC )(r, σ)  = dγC (r, σ) + PC (γ + γC )(r, σ) − dγC (r, σ) + PC (γC )(r, σ) = PC (γ + γC )(r, σ) − PC (γC )(r, σ).

(12)

Noting that PC is a function of three variables x, y and z, we see that Z 1 d PC (γ + γC ) − PC (γC ) = PC (tγ + γC ) dt dt 0 Z 1 ∂PC ∂PC (tγ + γC ) + ∇C γ · (tγ + γC ) dt. (13) = γ· ∂y ∂z 0 By Taylor’s Theorem, PC (γ+γC ) = PC (γC )+γ·

∂PC ∂PC (γC )+∇C γ· (γC )+O(r−2 |γ|2 +|∇C γ|2 ) (14) ∂y ∂z

when r−1 |γ| and |∇C γ| are small. Since dF |0 (α) = dα, as noted in Definition 4.6, dFC |γC (γ + γC ) = dγ and hence dPC |γC = 0. Thus, the first derivatives of PC with respect to y and z must vanish at γC by (14). Therefore, given small ǫ > 0, there exists a constant A0 > 0 such that ∂PC −2 (tγ + γ ) |γ| + r−1 |∇C γ|) and (15) C ≤ A0 (r ∂y ∂PC −1 (tγ + γ ) |γ| + |∇C γ|) (16) C ≤ A0 (r ∂z 12

for t ∈ [0, 1], whenever r−1 |γ|, r−1 |γC |, |∇C γ| and |∇C γC | ≤ ǫ.

(17)

The factors of r are determined by considering the scaling properties of γ and PC and their derivatives under changes in r. By (5), r−1 |γC | and |∇C γC | tend to zero as r → ∞. We can thus ensure that (17) is satisfied by the γC terms by making R larger. Hence, (17) holds if kγkC11 ≤ ǫ. Therefore, putting estimates (15) and (16) in (13) and using (12),  PF (α) Ψ(r, σ) = |PC (γ + γC )(r, σ) − PC (γC )(r, σ)| 2 ≤ A0 r−1 |γ(r, σ)| + |∇C γ(r, σ)|

(18)

for all r > R, σ ∈ Σ, whenever kγkC11 ≤ ǫ. Therefore, if α ∈ Cµ1 (U ), the corresponding γ lies in Cµ1 (Λ2+ T ∗ P ). Hence, by (18), PF (α) ∈ Cµ0 and there exists a constant c0 such that sup |ρ2−2µ PF (α)| ≤ c0 N

1 X

!2

sup |ρi−µ ∇i α|

i=0 N

whenever kαkC11 is sufficiently small, where ρ is a radius function on N . Thus, (8) holds for n = 0. Similar calculations give analogous results to (18) for derivatives of PF , from which we can deduce (8) for n > 0. We shall explain the method by considering 2 the first derivative. If γ ∈ Cloc (Λ2+ T ∗ P ), we calculate from (13):  ∇C PC (γ + γC ) − PC (γC )   Z 1 ∂PC ∂PC = ∇C γ · (tγ + γC ) + ∇C γ · (tγ + γC ) dt ∂y ∂z 0   Z 1 ∂PC ∂ 2 PC ∂ 2 PC 2 = ∇C γ · + γ · ∇C (tγ + γC ) · + ∇ (tγ + γ ) · C C ∂y ∂y 2 ∂y∂z 0   ∂ 2 PC ∂ 2 PC ∂PC + ∇C γ · ∇C (tγ + γC ) · + ∇2C (tγ + γC ) · + ∇2C γ · dt. ∂z ∂z∂y ∂z 2 Whenever kγkC11 ≤ ǫ there exists a constant A1 > 0 such that (15) and (16) hold with A0 replaced by A1 and, for t ∈ [0, 1], 2 2 2 ∂ PC ∂ PC ∂ PC ∂y 2 (tγ + γC ) , ∂y∂z (tγ + γC ) and ∂z 2 (tγ + γC ) ≤ A1 , 13

since the second derivatives of PC are continuous functions defined on the closed bounded set given by kγkC11 ≤ ǫ. We see that    ∇ PF (α) Ψ(r, σ) = ∇C PC (γ + γC ) − PC (γC ) (r, σ) ≤ A1 r

2 X i=0

ri−2 ∇iC γ(r, σ)

!2

whenever kγkC11 ≤ ǫ. From this we can deduce the result (8) for n = 1. In general we have the estimate !2 j+1 X j   ∇ PF (α) Ψ(r, σ) ≤ Aj rj ri−(j+1) ∇iC γ(r, σ) i=0

for some Aj > 0 whenever kγkC11 ≤ ǫ. The result (8) for all n ∈ N follows.



We now turn to the associated map G. Definition 4.9 Use the notation of Proposition 4.5 and Definition 4.6. Define 1 1 0 G : Cloc (U ) × Cloc (Λ4 T ∗ N ) → Cloc (Λ3 T ∗ N ) by: G(α, β) = F (α) + d∗ β. By the observations made in Definition 4.6, dG|(0,0) (α, β) = dα + d∗ β 1 (Λ2+ T ∗ N ⊕ Λ4 T ∗ N ). Thus G is a nonlinear elliptic operator for all (α, β) ∈ Cloc at (0, 0); that is, the linearisation of G at (0, 0) is elliptic.

To complete this subsection, we relate the kernels of F and G. Proposition 4.10 Use the notation of Definitions 3.1-3.4, 4.6 and 4.9. Let p > 4, k ≥ 1, l ∈ N, a ∈ (0, 1), µ < 0 and ν ≤ 0. The kernels of F and G in Cµl+1 , Cµl+1, a and Lpk+1, ν respectively are isomorphic. Proof: Note that ϕ is exact on any deformation Nα of N because it is closed and vanishes on N . Thus, if G(α, β) = 0,   d G(α, β) = d F (α) + dd∗ β = ∆β = 0,

since F (α) is exact. Recall we have a radius function ρ on N . If β decays with order O(ρµ ), or o(ρν ), as ρ → ∞, then ∗β is a harmonic function on N which tends to zero as ρ → ∞. The Maximum Principle allows us to deduce that ∗β = 0 and conclude that β = 0, from which the proposition follows.  14

4.3

Uniformly elliptic AC operators and regularity

We want to consider the regularity of solutions F (α) = 0 near 0, but this equation is not elliptic. Therefore, we study G(α, β) = 0 near (0, 0), which is a nonlinear elliptic equation. To do so, we must first discuss the regularity theory of certain linear elliptic operators, which we shall define, acting between weighted Banach spaces. We shall use the theory in [15] and [16], which centres around asymptotically cylindrical manifolds, so we begin with their definition. Definition 4.11 Recall Definition 3.1 and the notation introduced at the start of §4. In particular, we have a compact subset K of N such that N∞ = N \ K ∼ = (R, ∞) × Σ and a radius function ρ on N . Let g be the metric on N , considered as a Riemannian manifold. Define a metric g˜ on N by g˜ = ρ−2 g. Further, define a coordinate t on (T, ∞), where T = log R, by t = log r, let gcyl = dt2 + gΣ be ˜ be the Levi–Civita connection the cylindrical metric on (T, ∞) × Σ and let ∇ of g˜. We say that (N, g˜) is asymptotically cylindrical (ACyl). ˜ p (Λm T ∗ N ) to For p ≥ 1, k ∈ N and µ ∈ R we define the Banach space L k, µ be the subspace of Lpk, loc (Λm T ∗ N ) such that the following norm is finite:

kξkL˜ p

k µ

  p1 k Z X ˜ j ξ|p dVg˜  . = |ρ−µ ∇ j=0

N

Definition 4.12 Use the notation from Definition 4.11. Suppose that P : ′ ′ l 0 l 0 Cloc (Λm T ∗ N ) → Cloc (Λm T ∗ N ) and P∞ : Cloc (Λm T ∗ N∞ ) → Cloc (Λm T ∗ N∞ ) are linear differential operators of order l. Suppose further that P∞ is invariant under the R+ -action on N∞ ∼ = (T, ∞) × Σ. We say that P∞ is cylindrical. l For ξ ∈ Cloc (Λm T ∗ N∞ ), Pξ =

l X i=0

˜ iξ Pi · ∇

and P∞ ξ =

l X i=0

˜ i ξ, Pi, ∞ · ∇

where Pi and Pi, ∞ are tensors on N∞ of type (m + i, m′ ) and “ · ” means tensor product followed by contraction. If, for i = 0, . . . , l, ˜ j (Pi − Pi, ∞ )| → 0 |∇

for j ∈ N as t → ∞,

where | . | is calculated using gcyl , we say that P is asymptotically cylindrical (to m ∗ ˜p P∞ ). By [15, Theorem 3.7], P is a continuous map from L k+l, µ (Λ T N ) to ′ p m ∗ ˜ (Λ T N ) for all p > 1, k ∈ N and µ ∈ R. L k, µ 15

Definition 4.13 Use the notation from Definitions 4.11 and 4.12, so we have an operator P of order l acting between m- and m′ -forms on N . Let ν ∈ R. We say that P is asymptotically conical (AC) with rate ν if the differential operator ′

P ν = ρ−m +ν Pρm is asymptotically cylindrical to a cylindrical map P∞ , say. By [15, Proposition ′ and Definition 4.4], P : Lpk+l, µ (Λm T ∗ N ) → Lpk, µ−ν (Λm T ∗ N ) is continuous for all p > 1, k ∈ N and µ ∈ R. Note This definition can clearly be extended to linear differential operators acting between more general bundles of forms. Notice that an AC operator with rate ν reduces the growth rate of a form on the ends of N by ν. Examples of AC operators abound: d and d∗ are first-order operators with rate 1 and the Laplacian is a second-order operator with rate 2. Definition 4.14 Use the notation of Definition 4.13. We say that an AC operator P is uniformly elliptic if it is elliptic and P∞ is elliptic. Remark The definition of uniformly elliptic above implies uniform ellipticity in the sense of global bounds on the coefficients of the symbol. The operators d + d∗ and the Laplacian are uniformly elliptic AC operators. We now turn to regularity results for smooth uniformly elliptic AC operators. Theorem 4.15 Let V and W be bundles of forms on N and let P be a smooth uniformly elliptic AC operator from V to W of order l and rate ν, in the sense of Definitions 4.13 and 4.14. Let p > 1, k ∈ N, a ∈ (0, 1) and µ ∈ R. (a) Suppose that Pξ = η holds for ξ ∈ L1l, loc (V ) and η ∈ L10, loc (W ). If ξ ∈ Lp0, µ (V ) and η ∈ Lpk, µ−ν (W ), then ξ ∈ Lpk+l, µ (V ) and   kξkLpk+l, µ ≤ c kηkLpk, µ−ν + kξkLp0, µ for some constant c > 0 independent of ξ and η.

l 0 (b) Suppose that Pξ = η holds for ξ ∈ Cloc (V ) and η ∈ Cloc (W ). If ξ ∈ Cµ0 (V ) k, a and η ∈ Cµ−ν (W ), then ξ ∈ Cµk+l, a (V ) and   kξkCµk+l, a ≤ c′ kηkC k, a + kξkCµ0 µ−ν

for some constant c′ > 0 independent of ξ and η. Moreover, these estik, a mates hold if the coefficients of P only lie in Cloc . 16

These results may be deduced from those given in [21, §6.1.1] or [22]. By taking η = 0 in Theorem 4.15(b), we have the following useful corollary. Corollary 4.16 In the notation of Theorem 4.15, if ξ ∈ Cµl (V ) satisfies Pξ = 0 then ξ ∈ Cµ∞ (V ). We conclude by achieving the aim of this subsection. Proposition 4.17 Use the notation of Definition 4.9. Let (α, β) ∈ Lpk+1, µ (U )× Lpk+1, µ (Λ4 T ∗ N ) for p > 4, k ≥ 2 and µ < 1. If G(α, β) = 0 and kαkC11 is sufficiently small, (α, β) ∈ Cµ∞ (U ) × Cµ∞ (Λ4 T ∗ N ). Proof: Notice first that α and β lie in Cµ2 by Theorem 3.5, since k + 1 − 4p > 2. As noted in the proof of Proposition 4.10, G(α, β) = 0 implies that ∆β = 0. Hence, by Corollary 4.16, β ∈ Cµ∞ (Λ4 T ∗ N ). For the following argument we find it useful to work with weighted H¨ older spaces, defined in Definition 3.4. By Theorem 3.5, α ∈ Cµk, a (U ) with a = 1 − 4/p ∈ (0, 1) since p > 4. Let πΛ2+ be the projection from 2-forms to self-dual   2-forms on N . We know that d∗ G(α, β) = d∗ F (α) = 0 and also that   F˜ (α) = πΛ2+ d∗ F (α) = 0 is a nonlinear elliptic equation at 0, meaning that dF˜ |0 is an elliptic operator. We can write F˜ as   F˜ (α)(x) = RF x, α(x), ∇α(x) ∇2 α(x) + EF x, α(x), ∇α(x) ,

where RF and EF are smooth functions of their arguments, since F˜ (α) is linear in ∇2 α with coefficents depending on α and ∇α. Define  Sα (γ)(x) = RF x, α(x), ∇α(x) ∇2 γ(x)

2 for γ ∈ Cloc (Λ2+ T ∗ N ). Note that Sα is not the linearisation of F˜ . Then Sα is a linear uniformly elliptic second order AC operator with rate 2, if kαkC11 is sufficiently small, whose coefficients depend on x, α(x) and ∇α(x). These k−1, a coefficients therefore lie in Cloc . Recall the notation and results of Proposition 4.8 and that d∗ is an AC  operator with rate 1 in the sense of Definition 4.13. Thus, d∗ dα+d∗ PF (α) = 0  k−2, a and d∗ PF (α) ∈ C2µ−3 (Λ2 T ∗ N ). Therefore,

 k−2, a k−2, a Sα (α)(x) = −EF x, α(x), ∇α(x) ∈ C2µ−3 (Λ2+ T ∗ N ) ⊆ Cµ−2 (Λ2+ T ∗ N ), 17

 since µ < 1. However, EF x, α(x), ∇α(x) only depends on α and ∇α, and is at worst quadratic in these quantities by Proposition 4.8, so it must in fact k−1, a lie in Cµ−2 (Λ2+ T ∗ N ) since we are given control on the decay of the first k derivatives of α at infinity. k−1, a As α ∈ Cµ2 (Λ2+ T ∗ N ) and Sα (α) ∈ Cµ−2 (Λ2+ T ∗ N ), Theorem 4.15(b) implies k+1, a 2 ∗ that α ∈ Cµ (Λ+ T N ). Therefore we have proved α ∈ Cµk+1, a (Λ2+ T ∗ N ) only knowing a priori that α ∈ Cµk, a (Λ2+ T ∗ N ). We proceed by induction to show that α ∈ Cµl, a (Λ2+ T ∗ N ) for all l ≥ 2. The result follows from the elementary observation made at the end of Definition 3.4.  Taking β = 0 in Proposition 4.17 gives our main regularity result. Corollary 4.18 Use the notation of Definition 4.6. Let α ∈ Lpk+1, µ (U ) for p > 4, k ≥ 2 and µ < 1. If F (α) = 0 and kαkC11 is sufficiently small, α ∈ Cµ∞ (U ).

5

The map d + d∗ and the exceptional set D

We begin by defining the map of interest. Definition 5.1 Let p > 4, k ≥ 2 and µ < 1. Define the linear elliptic operator (d + d∗ )µ : Lpk+1, µ (Λ2+ T ∗ N ⊕ Λ4 T ∗ N ) → Lpk, µ−1 (Λ3 T ∗ N )

(19)

by (d + d∗ )µ (α, β) = dα + d∗ β. Let K(µ) = Ker(d + d∗ )µ . Remark The operator (d + d∗ )µ acts between the weighted Banach spaces claimed because it is AC with rate 1. We now make an important observation. Lemma 5.2 Use the notation from Proposition 4.5 and Definitions 4.9 and 5.1. By making the open set U smaller in C11 if necessary, G : Lpk+1, µ (U ) × Lpk+1, µ (Λ4 T ∗ N ) → Lpk, µ (Λ3 T ∗ N ).

(20)

Moreover, the linearisation of (20) at (0, 0) acts as (d + d∗ )µ , as in (19). Proof: By Proposition 4.8, and the fact that d and d∗ are AC operators with rate 1 on N , we see that G maps (α, β) ∈ Lpk+1, µ (U ) × Lpk+1, µ (Λ4 T ∗ N ) into Lpk, µ−1 (Λ3 T ∗ N ) if kαkC11 is sufficently small. This bound on the norm of α can be ensured by making the C11 -open set U smaller. The description of the linearisation follows from the observations in Definition 4.9.  18

It is clear from Propositions 4.7 and 4.10 that the kernel of the nonlinear map (20), for µ = λ, is intimately linked with the moduli space M(N, λ). Therefore, locally, one would expect the kernel K(λ) of the linear map (d + d∗ )λ , given by (19), to be related to M(N, λ) as well by Lemma 5.2. In fact, we shall see that K(λ) is directly connected with the infinitesimal deformations of N .

5.1

Fredholm and index theory

We want to understand the Fredholm theory of (19) and so state a result adapted from [16, Theorem 1.1 & Theorem 6.1]. Theorem 5.3 Let V and W be bundles of forms over N , let p > 1, let µ, ν ∈ R and let k, l ∈ N. Let P : Lpk+l, µ (V ) → Lpk, µ−ν (W ) be a uniformly elliptic AC operator of order l and rate ν in the sense of Definitions 4.13-4.14. There exists a countable discrete set D(P) ⊆ R, depending only on P∞ as in Definition 4.13, such that P is Fredholm if and only if µ ∈ / D(P). From this we know that, for each m ∈ N with m ≤ 4, there exists a countable discrete subset D(∆m ) of R such that the Laplacian on m-forms ∆m : Lpk+2, µ+1 (Λm T ∗ N ) → Lpk, µ−1 (Λm T ∗ N ) is Fredholm if and only if µ + 1 ∈ / D(∆m ). Thus, (d + d∗ )µ is Fredholm if  µ∈ / D(∆2 ) ∪ D(∆4 ) . However, we can give an explicit description of the set D for which (19) is not Fredholm, following [21, §6.1.2]. Recall the notation from Definitions 4.11-4.13 and consider the maps d and d acting on m-forms on N . These are asymptotically conical with rate 1, so the related asymptotically cylindrical operators d1 and (d∗ )1 are given by ∗

d1 = ρ−m dρm

and (d∗ )1 = ρ−m+2 d∗ ρm .

Notice that ρ is asymptotic to r = et , where the cylindrical coordinates (t, σ) on the ends N∞ ∼ = (T, ∞) × Σ of N were introduced in Definition 4.11. Thus, we can take the cylindrical operator (d + d∗ )∞ associated to d + d∗ to be (d + d∗ )∞ = e−mt (d + e2t d∗ )emt ,

(21)

acting on m-forms on the ends of N . Since N∞ ∼ = (T, ∞) × Σ, an m-form α on N∞ can be written as α(t, σ) = β(t, σ) + dt ∧ γ(t, σ), where, for each fixed t ∈ (T, ∞), β(t, σ) and γ(t, σ) are m- and (m− 1)-forms on Σ respectively. Therefore, if π : (0, ∞) × Σ → Σ is the natural projection, 19

∼ π ∗ (Λm T ∗ Σ) ⊕ π ∗ (Λm−1 T ∗ Σ). Hence, (d + d∗ )∞ maps sections of Λm T ∗ N∞ = π ∗ (Λ2 T ∗ Σ) ⊕ π ∗ (Λodd T ∗ Σ) to sections of π ∗ (Λodd T ∗ Σ) ⊕ π ∗ (Λeven T ∗ Σ). Moreover, this action is given by: ! α(t, σ) ∗ = (d + d )∞ β(t, σ) + γ(t, σ) ! ! ∂ dΣ + d∗Σ −( ∂t α(t, σ) + 3 − m) , (22) ∂ −(dΣ + d∗Σ ) β(t, σ) + γ(t, σ) ∂t + m where m denotes the operator which multiplies m-forms by a factor m, and dΣ and d∗Σ are the exterior derivative and its formal adjoint on Σ. However, we wish only to consider elements of Λ1 T ∗ Σ ⊕ Λ2 T ∗ Σ which correspond, via π ∗ , to self-dual 2-forms on N∞ . Thus we define VΣ ⊆ Λ2 T ∗ Σ ⊕ Λodd T ∗ Σ by VΣ = {(α, ∗Σ α + β) : α ∈ Λ2 T ∗ Σ, β ∈ Λ3 T ∗ Σ},

(23)

where ∗Σ is the Hodge star on Σ. Then π ∗ (VΣ ) ∼ = Λ2+ T ∗ N∞ ⊕ Λ4 T ∗ N∞ . We also want to project the image of (d + d∗ )∞ to 3-forms on N∞ , so we let WΣ = {(β, α) : α ∈ Λ2 T ∗ Σ, β ∈ Λ3 T ∗ Σ} and let πWΣ be the projection to WΣ . Note that π ∗ (WΣ ) ∼ = Λ3 T ∗ N∞ . 1 0 For w ∈ C, define a map (d + d∗ )∞ (w) : Cloc (VΣ ⊗ C) → Cloc (WΣ ⊗ C) by: ! α(σ) (d + d∗ )∞ (w) = ∗Σ α(σ) + β(σ) ! ! dΣ + d∗Σ −(w + 3 − m) α(σ) . (24) πWΣ ◦ w+m −(dΣ + d∗Σ ) ∗Σ α(σ) + β(σ) ∂ in (22). Notice that we have formally substituted w for ∂t p Let p > 4 and k ≥ 2 as in (19), noting that Lk+1 ֒→ C 2 on Σ by the Sobolev Embedding Theorem. Define C ⊆ C as the set of w for which the map

(d + d∗ )∞ (w) : Lpk+1 (VΣ ⊗ C) → Lpk (WΣ ⊗ C)

(25)

is not an isomorphism. By the proof of [16, Theorem 1.1], D = {Re w : w ∈ C}. In fact, C ⊆ R by [21, Lemma 6.1.13], which shows that the corresponding sets C(∆m ) are all real. Hence C = D. The symbol, hence the index indw , of (d + d∗ )∞ (w) is independent of w. Furthermore, (d + d∗ )∞ (w) is an isomorphism for generic values of w since D is countable and discrete. Therefore indw = 0 for all w ∈ C; that is, dim Ker(d + d∗ )∞ (w) = dim Coker(d + d∗ )∞ (w), 20

so that (25) is not an isomorphism precisely when it is not injective. The condition (d + d∗ )∞ (w) = 0, using (24) and elliptic regularity, corresponds to the existence of α ∈ C ∞ (Λ2 T ∗ Σ) and β ∈ C ∞ (Λ3 T ∗ Σ) satisfying dΣ α = wβ

and dΣ ∗Σ α + d∗Σ β = (w + 2)α.

(26)

We first note that (26) implies that dΣ d∗Σ β = ∆Σ β = w(w + 2)β.

(27)

Since eigenvalues of the Laplacian on Σ are positive, β = 0 if w ∈ (−2, 0). If w = 0 and we take α = 0, (26) forces β to be coclosed. As there are nontrivial coclosed 3-forms on Σ, (d + d∗ )∞ (0) is not injective and hence 0 ∈ D. Suppose 3 that w = −2 lies in D. Then (26) gives [β] = 0 in HdR (Σ). We know that β is harmonic from (27) so, by Hodge theory, β = 0. Therefore −2 ∈ D if and only if there exists a nonzero closed and coclosed 2-form on Σ. We state a proposition which follows from the work above. Proposition 5.4 Recall the definition of Σ from the start of §4 and denote the Hodge star, the exterior derivative and its formal adjoint on Σ by ∗Σ , dΣ and d∗Σ respectively. Let D(µ) = {(α, β) ∈ C ∞ (Λ2 T ∗ Σ⊕Λ3 T ∗ Σ) : dΣ α = µβ, dΣ ∗Σ α+d∗Σ β = (µ+2)α}. The countable discrete set D of µ ∈ R for which (d + d∗ )µ , given in (19), is not Fredholm is given by: D = {µ ∈ R : D(µ) 6= 0}. Moreover, −2 ∈ D if and only if b1 (Σ) > 0, and 0 ∈ D.

A perhaps more illuminating way to characterise D(µ) is by: (α, β) ∈ D(µ) ⇐⇒

ξ = (rµ+2 α + rµ+1 dr ∧ ∗Σ α, rµ+3 dr ∧ β) ∈ C ∞ (Λ2+ T ∗ C ⊕ Λ4 T ∗ C) is an O(rµ ) solution of (d + d∗ )ξ = 0 in C ∞ (Λ3 T ∗ C) as r → ∞.

We now make a definition as in [16]. Definition 5.5 Recall the map (d + d∗ )∞ defined by (22), the bundle VΣ given in (23) and the set D given in Proposition 5.4. Let µ ∈ D. We define d(µ) to be the dimension of the vector space of solutions of (d + d∗ )∞ ξ = 0 of the form ξ(t, σ) = eµt p (t, σ), where p (t, σ) is a polynomial in t ∈ (T, ∞) with coefficients in C ∞ (VΣ ⊗ C). The next result is immediate from [16, Theorem 1.2]. 21

Proposition 5.6 Use the notation of Proposition 5.4 and Definition 5.5. Let λ, λ′ ∈ / D with λ′ ≤ λ. For any µ ∈ / D let iµ (d + d∗ ) denote the Fredholm index of the map (19). Then X d(µ). iλ (d + d∗ ) − iλ′ (d + d∗ ) = µ∈D ∩(λ′ , λ)

Note A similar result holds for any uniformly elliptic AC operator on N . We make a key observation, which shall be used on a number of occasions later. Proposition 5.7 Use the notation of Theorem 5.3. Let λ, λ′ ∈ R be such that λ′ ≤ λ and [λ′ , λ] ∩ D(P) = ∅. The kernels, and cokernels, of P : Lpk+l, µ (V ) → Lpk, µ−ν (W ) when µ = λ and µ = λ′ are equal. Proof: Denote the dimensions of the kernel and cokernel of P : Lpk+l, µ (V ) → Lpk, µ−ν (W ), for µ ∈ / D(P), by k(µ) and c(µ) respectively. Notice that these dimensions are finite since P is Fredholm if µ ∈ / D(P). Since [λ′ , λ] ∩ D(P) = ∅, k(λ) − c(λ) = k(λ′ ) − c(λ′ ) by [16, Theorem 1.2] and hence k(λ) − k(λ′ ) = c(λ) − c(λ′ ).

(28)

We know that k(λ) ≥ k(λ′ ) because Lpk+1, λ′ ֒→ Lpk+1, λ by Theorem 3.5(a) as λ ≥ λ′ . Similarly, since c(µ) is equal to the dimension of the kernel of the formal adjoint operator acting on the dual Sobolev space with weight −4 − (µ − ν) (as noted in Definition 3.2), c(λ) ≤ c(λ′ ). Since the left-hand side of (28) is nonnegative and the right-hand side is non-positive, we conclude that both must be zero. The result follows from the fact that the kernel of P in Lpk+1, λ′ is contained in the kernel of P in Lpk+1, λ , and vice versa for the cokernels.  We conclude with an explicit description of the quantity d(µ) for µ ∈ D. This result, as can be seen from the proof, is similar to [5, Proposition 2.4]. Proposition 5.8 In the notation of Proposition 5.4 and Definition 5.5, d(µ) = dim D(µ) for µ ∈ D. Proof: Use the notation of Proposition 5.4 and the work preceding it and Definition 5.5. Let p(t, σ) be a polynomial in t ∈ (T, ∞) of degree m written as   m m X X  ∗Σ pj (σ) + qj (σ) tj  , pj (σ)tj , p(t, σ) =  j=0

j=0

22

where pj ∈ C ∞ (Λ2 T ∗ Σ) and qj ∈ C ∞ (Λ3 T ∗ Σ) for j = 0, . . . , m, with pm and qm not both zero, and let ξ(t, σ) = eµt p(t, σ) as in Definition 5.5. We want to find the dimension d(µ) of the space of ξ such that (d + d∗ )∞ ξ = 0. Using (22), (d + d∗ )∞ ξ = 0 is equivalent to m X j=0

m X j=0

tj (dΣ pj − µqj ) −

m X

jtj−1 qj = 0 and

(29)

j=0

m  X jtj−1 pj = 0. tj (µ + 2)pj − dΣ ∗Σ pj − d∗Σ qj +

(30)

j=0

Comparing coefficients of tm we deduce that (pm , qm ) ∈ D(µ). Suppose, for a contradication, that m ≥ 1. Comparing coefficients of tm−1 in (29) and (30): dΣ pm−1 − µqm−1 = mqm

and dΣ ∗Σ pm−1 + d∗Σ qm−1 − (µ + 2)pm−1 = mpm . (31)

We then compute using (31) and the fact that (pm , qm ) ∈ D(µ): mhpm , pm iL2 = hpm , dΣ ∗Σ pm−1 + d∗Σ qm−1 − (µ + 2)pm−1 iL2 = hdΣ ∗Σ pm − (µ + 2)pm , pm−1 iL2 + hdΣ pm , qm−1 iL2

= h−d∗Σ qm , pm−1 iL2 + hµqm , qm−1 iL2 = −hqm , dΣ pm−1 − µqm−1 iL2 = −mhqm , qm iL2 . Hence, m(kpm k2L2 + kqm k2L2 ) = 0 and so pm = qm = 0, our required contradiction. Thus, the solutions ξ must be of the from

 ξ(t, σ) = eµt p(t, σ) = eµt p0 (σ), ∗Σ p0 (σ) + q0 (σ)

for (p0 , q0 ) ∈ D(µ). The proposition follows.

5.2



The image of d + d∗

We remarked earlier upon the connection of the kernel of (d + d∗ )µ with the infinitesimal deformations of N . This suggests that the cokernel of (d + d∗ )µ is related to the obstruction theory for the deformation problem. When (d + d∗ )µ 23

is Fredholm, the cokernel is isomorphic, via the dual pairing given in Definition 3.2, to the kernel of the adjoint map which we now define. Definition 5.9 Let p > 4, k ≥ 2 and µ < 1. Let q > 1 be such that and let l ≥ 2. The adjoint map to (19) is given by

1 p

+

1 q

(d∗+ + d)µ : Lql+1, −3−µ (Λ3 T ∗ N ) → Lql, −4−µ (Λ2+ T ∗ N ⊕ Λ4 T ∗ N ),

=1

(32)

where (d∗+ +d)µ (γ) = d∗+ γ+dγ and d∗+ = 21 (d∗ +∗d∗ ). Let C+ (µ) = Ker(d∗+ +d)µ . It is straightforward to see, using integration by parts and the dual pairing, that (d∗+ + d)µ is the formal adjoint to (d + d∗ )µ . Remark The space C+ (µ) is the kernel of an elliptic map so, for µ ∈ / D, it is a finite-dimensional space of smooth forms by Corollary 4.16. Thus, C+ (µ) is independent of l and we can choose l ≥ 2 for use later. The aim of this subsection is to prove the following. Theorem 5.10 Use the notation of Proposition 5.4 and Definition 5.9. Suppose further that µ ∈ (−∞, 1) \ D and let C(µ) = {γ ∈ Lql+1, −3−µ (Λ3 T ∗ N ) : dγ = d∗ γ = 0}.

(33)

˜ There exist finite-dimensional subspaces C(µ) and O(N, µ) of Lpk, µ−1 (Λ3 T ∗ N ) such that    ˜ Lpk, µ−1 (Λ3 T ∗ N ) = d Lpk+1, µ (Λ2 T ∗ N ) + d∗ Lpk+1, µ (Λ4 T ∗ N ) ⊕ C(µ) (34)

and

  d Lpk+1, µ (Λ2 T ∗ N ) = d Lpk+1, µ (Λ2+ T ∗ N ) ⊕ O(N, µ).

(35)

∼ ˜ ˜ ⊕ O(N, µ) ∼ Moreover, C(µ) = C+ (µ) via the dual = C(µ) and C˜+ (µ) = C(µ) pairing. Furthermore: (a) if µ < −2, the sum in (34) is a direct sum; (b) if µ ∈ [−2, 0), O(N, µ) = 0 and the sum in (34) is a direct sum; (c) if µ ∈ [0, 1), O(N, µ) = 0 but the sum in (34) is not necessarily direct. Before proving the theorem we make an elementary observation.   Lemma 5.11 If µ < 0, d Lpk+1, µ (Λ2 T ∗ N ) ∩ d∗ Lpk+1, µ (Λ4 T ∗ N ) = {0}. 24

Proof: If β ∈ Lpk+1, µ (Λ4 T ∗ N ) such that d∗ β is exact, then dd∗ β = 0. Therefore ∗β is a harmonic function which is o(ρµ ) as ρ → ∞. Applying the Maximum Principle, ∗β = 0.  Proof of Theorem 5.10(a). The key to proving this part of the theorem lies with comparing the image of (d + d∗ )µ with the image of the map d + d∗ : Lpk+1, µ (Λ2 T ∗ N ⊕ Λ4 T ∗ N ) → Lpk, µ−1 (Λ3 T ∗ N )

(36)

given by (d + d∗ )(α, β) = dα + d∗ β. Clearly, there exists a finite-dimensional ˜ subspace C(µ) of Lpk, µ−1 (Λ3 T ∗ N ), which is isomorphic (via the dual pairing) to the annihilator of the image of (36), such that    ˜ Lpk, µ−1 (Λ3 T ∗ N ) = d Lpk+1, µ (Λ2 T ∗ N ) ⊕ d∗ Lpk+1, µ (Λ4 T ∗ N ) ⊕ C(µ). (37) (The direct sum follows from Lemma 5.11 as µ < −2.) Now, the annihilator A(µ) of the image of (36) is given by: A(µ) = {γ ∈ Lql+1, −3−µ (Λ3 T ∗ N ) : hγ, dα + d∗ βi = 0

for all (α, β) ∈ Lpk+1, µ (Λ2 T ∗ N ⊕ Λ4 T ∗ N )}.

Therefore, using integration by parts (justified by the choice of weight for the dual Sobolev space), we deduce that A(µ) = C(µ). Using the Fredholmness of (d + d∗ )µ , since µ ∈ / D, there exists a finitep 3 ∗ ˜ dimensional subspace C+ (µ) of Lk, µ−1 (Λ T N ) such that   Lpk, µ−1 (Λ3 T ∗ N ) = d Lpk+1, µ (Λ2+ T ∗ N ) ⊕ d∗ Lpk+1, µ (Λ4 T ∗ N ) ⊕ C˜+ (µ). (38)

Moreover, C˜+ (µ) is isomorphic to the annihilator of the image of (19), which is equal to C+ (µ) as given in Definition 5.9 (again using integration by parts).  Equation (38) allows us to deduce that d∗ Lpk+1, µ (Λ4 T ∗ N ) is closed. Hence, Lemma 5.11 allows us to deduce that   d Lpk+1, µ (Λ2 T ∗ N ) ∩ d∗ Lpk+1, µ (Λ4 T ∗ N ) = {0}.

Comparing (37) and (38) we see that there exists a finite-dimensional space ˜ ⊕ O(N, µ), from which (a) follows. O(N, µ) such that C˜+ (µ) = C(µ) 

We shall require some preliminary technical results before resuming our proof of Theorem 5.10. We consider the maps d : Lpk+1, µ (Λ2+ T ∗ N ) → Lpk, µ−1 (Λ3 T ∗ N ) and d:

Lpk+1, µ (Λ2 T ∗ N )



Lpk, µ−1 (Λ3 T ∗ N ).

25

(39) (40)

Proposition 5.12 Let p > 4, k ≥ 2 and µ ∈ (−∞, 1) \ D, where D is given in Proposition 5.4. Let the annihilators of the images of (39) and (40) be A+ (µ) and A(µ) respectively. Then, if q > 1 such that p1 + q1 = 1 and l ≥ 2, A+ (µ) = {γ ∈ Lql+1, −3−µ (Λ3 T ∗ N ) : d∗ γ ∈ Lql, −4−µ (Λ2− T ∗ N )}

(41)

and A(µ) = {γ ∈ Lql+1, −3−µ (Λ3 T ∗ N ) : d∗ γ = 0}.

(42)

Furthermore, if µ ∈ [−2, 1) \ D, A+ (µ) = A(µ). Proof: The formulae (41) and (42) are easily deduced using the dual pairing and integration by parts. Clearly A(µ) ⊆ A+ (µ), so suppose γ ∈ A+ (µ) and µ ≥ −2. Notice that p > 4 and p1 + q1 = 1 force q ∈ (1, 43 ), and µ ≥ −2 implies that −4 − µ ≤ −2. Therefore, by Theorem 3.5(a), Lql, −4−µ ֒→ L20, −2 = L2 , recalling (3). Thus, Z Z Z −d(∗γ ∧ d∗γ) = 0. −d∗γ ∧ d∗γ = −d∗ γ ∧ d∗ γ = kd∗ γk2L2 = N

N

N

−3−µ

The integration by parts is valid since ∗γ = o(ρ ) and d∗γ = o(ρ−4−µ ) as ρ → ∞, and −7 − 2µ ≤ −3. The proof is thus complete.  Using this proposition we can deduce the following invaluable result.

Proposition 5.13 Recall the definition of D in Proposition 5.4. For p > 4, k ≥ 2 and µ ∈ [−2, 1) \ D,       d Lpk+1, µ (Λ2 T ∗ N ) = d Lpk+1, µ (Λ2 T ∗ N ) = d Lpk+1, µ (Λ2+ T ∗ N ) .

Proof: By Proposition 5.12, the annihilators of the images of (39) and (40) are equal. We deduce that the closure of the ranges of (39) and (40) are equal. However, we know that µ ∈ / D so that (19) is Fredholm, which means that it has closed range. Therefore, (39) has closed range and hence     d Lpk+1, λ (Λ2+ T ∗ N ) = d Lpk+1, λ (Λ2 T ∗ N ) . The left-hand side of this equation is contained in the image of (40), whereas the right-hand side contains the image of (40). The result follows.  Proof of Theorem 5.10(b)-(c). If µ ∈ [−2, 1) \ D, (19) is Fredholm, so there ˜ exists a finite-dimensional subspace C(µ) of Lpk, µ−1 (Λ3 T ∗ N ) such that    ˜ Lpk, µ−1 (Λ3 T ∗ N ) = d Lpk+1, µ (Λ2+ T ∗ N ) + d∗ Lpk+1, µ (Λ4 T ∗ N ) ⊕ C(µ).

The results now follow from Lemma 5.11 and Proposition 5.13.

26



5.3

The kernel of the adjoint map

We already remarked that studying the kernel of the adjoint map will help us to understand the obstructions to deformations of N . For this subsection we use the notation of Definition 5.9 and Proposition 5.12. Suppose further that µ ∈ [−2, 1) \ D, where D is given in Proposition 5.4. Let γ ∈ A+ (µ). By Proposition 5.12, ∗γ ∈ Lql+1, −3−µ (Λ1 T ∗ N ) and satisfies d∗γ = 0. Recall that Ψ : (R, ∞) × Σ ∼ = N \ K and that (r, σ) are the coordinates on (R, ∞) × Σ. So, on (R, ∞) × Σ,  ∗γ Ψ(r, σ) = χ(r, σ) + f (r, σ) dr

for a function f and 1-form χ on (R, ∞) × Σ, where χ has no dr component. Write the exterior derivative on (R, ∞) × Σ in terms of the exterior derivative dΣ on Σ as: ∂ . d = dΣ + dr ∧ ∂r The equation d∗γ = 0 then implies that dΣ χ = 0

∂χ − dΣ f = 0. ∂r

and

(43)

Define a function ζ on (R, ∞) × Σ by ζ(r, σ) = −

Z



f (s, σ) ds.

r

This is well-defined since the modulus of f is o(r−3−µ ) as r → ∞, where −3−µ ≤ −1 since µ ≥ −2. Noting that the modulus of χ with respect to gΣ is o(r−2−µ ) as r → ∞, with −2 − µ ≤ 0, we calculate using (43): Z ∞ Z ∞ ∂χ (s, σ) ds + f (r, σ) dr dζ(r, σ) = − dΣ f (s, σ) ds + f (r, σ) dr = − ∂r r r h i∞  = − χ(s, σ) + f (r, σ) dr = χ(r, σ) + f (r, σ) dr = ∗γ Ψ(r, σ) . r

If {R} × Σ has a tubular neighbourhood in N , which can be ensured by making R larger if necessary, we can extend ζ smoothly to a function on N . Hence ζ ∈ Lql+2, −2−µ (Λ0 T ∗ N ) with dζ = ∗γ on N \ K. This leads us to the following.

Proposition 5.14 Use the notation of Definition 5.9 and Propositions 5.4 and 5.12. Let γ ∈ A+ (µ) for µ ∈ [−2, 1) \ D. There exists ζ ∈ Lql+2, −2−µ (Λ0 T ∗ N ) such that ∗γ − dζ = γˆ is a closed compactly supported 1-form. Moreover, the 1 map γ 7→ [ˆ γ ] from C+ (µ) ⊆ A+ (µ) to Hcs (N ) is injective. 27

Proof: Clearly our construction above ensures that γˆ is a closed 1-form which is 1 zero outside of the compact subset K of N . Thus [ξ] ∈ Hcs (N ). Suppose that γ ∈ C+ (µ) and [ˆ γ ] = 0. Then ξ = dζ˜ for some function ζ˜ with compact support. Therefore ˜ 0 = d∗ ∗γ = d∗ (dζ + γˆ ) = d∗ d(ζ + ζ). Hence ζ + ζ˜ is a harmonic function of order o(ρ−2−µ ) as ρ → ∞. Since µ ≥ −2, the Maximum Principle forces ζ + ζ˜ = 0 and the result follows.  1 It follows from Proposition 7.6 below that the map from C+ (µ) to Hcs (N ) is an isomorphism when µ ∈ [−2, 0) \ D.

6

The deformation theory

In this section we prove our main result, which is a local description of the moduli space M(N, λ) of AC coassociative deformations of N with rate λ. We remind the reader that we are using notation from the start of §4. We shall further assume that λ ∈ / D, where D is given by Proposition 5.4, and we choose some p > 4 and k ≥ 2.

6.1

Deformations and obstructions

We begin by identifying the infinitesimal deformation space for our moduli space problem. Definition 6.1 The infinitesimal deformation space is I(N, λ) = {α ∈ Lpk+1, λ (Λ2+ T ∗ N ) : dα = 0}. Note that I(N, λ) is a subspace of K(λ), given in Definition 5.1, and is thus finite-dimensional as λ ∈ / D. In fact, I(N, λ) ∼ = K(λ) if λ < 0 by Lemma 5.11. We say that our deformation theory is unobstructed if M(N, λ), given in Definition 4.1, is a smooth manifold near N of dimension dim I(N, λ). By Proposition 4.7 and Corollary 4.18, M(N, λ) is homeomorphic near N to the kernel of the map F , given in Definition 4.6, near 0 in Lpk+1, λ (U ). Clearly I(N, λ) is the tangent space to Ker F at 0, so it can be identified with the infinitesimal deformation space. Furthermore, if there are no obstructions to the deformation theory of N , then every infinitesimal deformation should extend to a genuine deformation and the moduli space should be a smooth manifold. This justifies our definition of an unobstructed deformation theory. 28

Theorem 5.10 identifies the obstruction space. Definition 6.2 The obstruction space is the finite-dimensional space  d Lpk+1, λ (Λ2 T ∗ N ) ∼  O(N, λ) = d Lpk+1, λ (Λ2+ T ∗ N ) given in (35). We observe that

dim O(N, λ) = dim C+ (λ) − dim C(λ), in the notation of Definition 5.9 and Theorem 5.10. It shall become clear in the next subsection why we should think of O(N, λ) as the obstructions to our deformation theory. However, we notice by Theorem 5.10 that O(N, λ) = {0} for λ ∈ [−2, 1), which should mean that our deformation theory is unobstructed for these rates – we shall confirm that this is the case. The key step in understanding the obstruction theory is contained in the next result, which studies the image of the deformation map F . Proposition 6.3 Use the notation of Proposition 4.5 and Definition 4.6. Making the open set U smaller if necessary,  F : Lpk+1, λ (U ) → d Lpk+1, λ (Λ2 T ∗ N ) ⊆ Lpk, λ−1 (Λ3 T ∗ N ). Proof: It was noted in the proof of Proposition 4.10 that F (α) is exact for α ∈ Lpk+1, λ (U ). However, we need to know that we can choose a 2-form H(α)  lying in an appropriate weighted Sobolev space such that d H(α) = F (α). Let u be the vector field given by dilations, which, in coordinates (x1 , . . . , x7 ) on R7 , is written: ∂ ∂ u = x1 + . . . + x7 . (44) ∂x1 ∂x7 Then the Lie derivative of ϕ along u is: Lu ϕ = d(u · ϕ) = 3ϕ. Therefore, ψ =

1 3

(45)

u · ϕ is a 2-form such that dψ = ϕ. Note that ψ|C ≡ 0 since (u · ϕ)|C = u · (ϕ|C ) = 0,

as u ∈ T C and C is coassociative by Proposition 2.8. Recall the definition of 1 fα and Nα in Definition 4.6. Define, for α ∈ Cloc (U ), H(α) = fα∗ ψ|Nα 29



so that     F (α) = fα∗ dψ|Nα = d fα∗ ψ|Nα = d H(α) .

Recall the diffeomorphism Ψα : (R, ∞)×Σ → Nα \Kα , where Kα is compact, introduced before Proposition 4.7, and the inclusion map ι : (R, ∞) × Σ → C given by ι(r, σ) = rσ. The decay of H(α) at infinity is determined by: Ψ∗α (ψ) = (Ψ∗α − ι∗ )(ψ) + ι∗ (ψ) = (Ψ∗α − ι∗ )(ψ) since ψ|C ≡ 0. For (r, σ) ∈ (R, ∞) × Σ,    (Ψ∗α − ι∗ )(ψ)|(r,σ) = dΨα |∗(r,σ) ψ|Ψα (r,σ) − dι|∗(r,σ) ψ|Ψα (r,σ)  + dι|∗(r,σ) ψ|Ψα (r,σ) − ψ|rσ ,

(46)

using the linearity of dι∗ to derive the last term. Since |ψ| = O(r) and Ψα satisfies (1) so that |dΨ∗α − dι∗ | = O(rλ−1 ) as r → ∞, the expression in brackets in (46) is O(rλ ). The magnitude of the final term in (46) at infinity is determined by the behaviour of dι∗ , ∇ψ and Ψα −ι. Hence, as |dι∗ | and |∇ψ| are O(1), using (1) again implies that this term is O(rλ ). We conclude that if α ∈ Lpk+1, λ (U ) then H(α) ∈ Lpk, λ (Λ2 T ∗ N ). Notice that H(α) has one degree of differentiability less than one would expect since it depends on α and ∇α. By Proposition 5.12, the annihilator A(λ) of the image of (40) for µ = λ comprises of coclosed forms and lies in Lql+1, −3−λ , where 1p + 1q = 1 and l ≥ 2. For γ ∈ A(λ), recalling the dual pairing given in Definition 3.2,  hF (α), γi = hd H(α) , γi = hH(α), d∗ γi = 0, where the integration by parts is valid as H(α) ∈ Lpk, λ and γ ∈ Lql+1, −3−λ . Thus, F (α) must lie in the closure of the image of (40). 

6.2

The moduli space M(N, λ)

We now state and prove our main result. Theorem 6.4 Let N be a coassociative 4-fold in R7 which is asymptotically conical with rate λ. Let p > 4, k ≥ 2 and suppose that λ ∈ / D, where D is given in Proposition 5.4. Use the notation of Definitions 4.1, 5.1, 6.1 and 6.2 and let   B(λ) = β ∈ Lpk+1, λ (Λ4 T ∗ N ) : d∗ β ∈ d Lpk+1, λ (Λ2 T ∗ N ) ⊆ Lpk, λ−1 (Λ3 T ∗ N ) .

ˆ There exist a manifold M(N, λ), which is an open neighbourhood of 0 in ˆ K(λ), and a smooth map π : M(N, λ) → O(N, λ), with π(0) = 0, such that an open neighbourhood of 0 in Ker π is homeomorphic to M(N, λ) near N . 30

Moreover, if O(N, λ) = {0}, M(N, λ) is a smooth manifold near N with dim M(N, λ) = dim I(N, λ) = dim K(λ) − dim B(λ); that is, the deformation theory is unobstructed. Note By Theorem 5.10, B(λ) = {0} if λ < 0 and, if λ ≥ 0,   B(λ) = β ∈ Lpk+1, λ (Λ4 T ∗ N ) : d∗ β ∈ d Lpk+1, λ (Λ2+ T ∗ N ) .

Moreover, B(λ) is finite-dimensional as it is isomorphic to a subspace of the harmonic functions in Lpk+1, λ . Proof: Recall the open set U given by Propostion 4.5. Make U smaller if necessary so that Proposition 6.3 holds and that the C11 -norm of elements α ∈ Lpk+1, λ (U ) is small enough for Corollary 4.18 to apply. Define V = Lpk+1, λ (U ) × Lpk+1, λ (Λ4 T ∗ N )

X = Lpk+1, λ (Λ2+ T ∗ N ) ⊕ Lpk+1, λ (Λ4 T ∗ N )

Y = O(N, λ) ⊆ Lpk, λ−1 (Λ3 T ∗ N ) and   Z = d Lpk+1, λ (Λ2 T ∗ N ) + d∗ Lpk+1, λ (Λ4 T ∗ N ) ⊆ Lpk, λ−1 (Λ3 T ∗ N ).

By Definition 3.1, X is a Banach space and, as noted at the start of §4.2, V is an open neighbourhood of zero in X. The obstruction space Y is trivially a Banach space since it is finite-dimensional. Finally, Z is a Banach space as it is a closed subset of a Banach space by the proof of Theorem 5.10. Define a smooth map G on the open subset V × Y of X × Y by G(α, β, γ) = G(α, β) + γ, where G is given in Definition 4.9. By Proposition 6.3, G maps into Z. Moreover, the linearisation of G at zero, dG|(0,0,0) : X × Y → Z, acts as (α, β, γ) 7−→ dα + d∗ β + γ. Therefore, dG|(0,0,0) is surjective by Theorem 5.10. Furthermore, using the notation of Definition 5.1, we see that Ker dG|(0,0,0) = {(α, β, γ) ∈ X × Y : dα + d∗ β + γ = 0} ∼ = Ker(d + d∗ )λ = K(λ)

31

since, by definition of Y , (d + d∗ )λ (X) ∩ Y = {0}. Finally note that there exists a closed subspace A of X such that X = K(λ) ⊕ A as K(λ) is the kernel of a Fredholm operator. We now apply the Implicit Function Theorem (Theorem 3.6) to G. Thus, ˆ there exist open sets M(N, λ) ⊆ K(λ), WA ⊆ A and WY ⊆ Y , each containing ˆ ˆ zero, with M(N, λ) × WA ⊆ V , and smooth maps WA : M(N, λ) → WA and ˆ WY : M(N, λ) → WY such that  ˆ Ker G ∩ M(N, λ) × WA × WY   ˆ = ξ, WA (ξ), WY (ξ) ∈ K(λ) ⊕ A ⊕ Y : ξ ∈ M(N, λ) .

We take our map π = WY . Thus, an open neighbourhood of zero in Ker π is homeomorphic to an open neighbourhood of zero in Ker G ⊆ V . If λ < 0, Proposition 4.10 and Theorem 5.10(a)-(b) imply that Ker G ⊆ V is isomorphic to Ker F ⊆ Lpk+1, λ (U ), I(N, λ) ∼ = K(λ) and B(λ) = {0}. Since Ker F consists of smooth forms by Corollary 4.18 and gives a local description of the moduli space by Proposition 4.7, the result follows for λ < 0. Therefore, suppose λ ≥ 0. Define a smooth map πG on Ker G by πG (α, β) = β. Notice that p Ker πG ∼ = Ker F = {α ∈ Lk+1, λ (U ) : F (α) = 0}.

Moreover, if (α, β) ∈ Ker G,  d∗ β = −F (α) ∈ d Lpk+1, λ (Λ2 T ∗ N )

by Proposition 6.3. Therefore, πG : Ker G → B(λ) and dπG |(0,0) : K(λ) → B(λ). Since dπG |(0,0) is clearly surjective, πG is locally surjective. Moreover, p Ker dπG |(0,0) ∼ = {α ∈ Lk+1, λ (Λ2+ T ∗ N ) : dα = 0} = I(N, λ) = Ker dF |0 .

Hence, Ker F is locally diffeomorphic to Ker dπG |(0,0) and the result follows.  Theorem 5.10 gives us an immediate corollary to Theorem 6.4. Corollary 6.5 The deformation theory of AC coassociative 4-folds N in R7 with generic rate λ ∈ [−2, 1) is unobstructed. Remark We have no guarantee that the deformation theory is unobstructed for generic rates λ < −2. However, we would expect the obstruction space to be zero for generic choices of N .

32

7

The dimension of the moduli space M(N, λ)

We shall continue to use the notation from the start of §4. We additionally recall the notation of Definitions 5.1 and 5.9: for µ ∈ R, we denote the kernel of (d + d∗ )µ , given in (19), by K(µ) and the kernel of (d∗+ + d)µ , given in (32), by C+ (µ). We also remind the reader that C+ (µ) is isomorphic to the cokernel of (19) via the pairing given in Definition 3.2 when µ ∈ / D, where D is given in Proposition 5.4. We shall split our study of the dimension of the moduli space into three ranges of rates λ ∈ / D: λ ∈ [−2, 0), λ ∈ [0, 1) and λ < −2.

7.1

Topological calculations

The first proposition we state follows from standard results in algebraic topology if we consider our coassociative 4-fold N , which is AC to C ∼ = (0, ∞) × Σ, as the interior of a manifold which has boundary Σ. This result is invaluable for our dimension calculations. Proposition 7.1 Recall the notation from the start of §4. Let the map φm : m m Hcs (N ) → HdR (N ) be defined by φm ([ξ]) = [ξ]. Let r > R and let Ψr : Σ → N m m be the embedding given by Ψr (σ) = Ψ(r, σ). Define pm : HdR (N ) → HdR (Σ) ∞ ∗ by pm ([ξ]) = [Ψr ξ]. Let f ∈ C (N ) be such that f = 0 on K and f = 1 on (R + 1, ∞) × Σ. If πΣ : (R, ∞) × Σ ∼ = N \ K → Σ is the projection map, define ∗ m+1 m (N ) by ∂m ([ξ]) = [d(f πΣ ξ)]. Then the following sequence (Σ) → Hcs ∂m : HdR is exact: φm

pm



m m m m+1 m · · · −→ Hcs (N ) −→ HdR (N ) −→ HdR Hcs (N ) −→ · · · . (Σ) −→

(47)

0 4 Remark The fact that Hcs (N ) = HdR (N ) = {0} enables us to calculate the dimension of various spaces more easily using the long exact sequence (47).

We now identify a space of forms which we shall relate to the topology of N . Definition 7.2 Define Hm (N ) by Hm (N ) = {ξ ∈ L2 (Λm T ∗ N ) : dξ = d∗ ξ = 0}.

(48)

Using (3), we notice that L20, −2 = L2 . Moreover, the elliptic regularity result Theorem 4.15(a) applied to the Laplacian can be improved as in [15, Proposition 5.3], using more theory of weighted Sobolev spaces than we wish to discuss here, to show that harmonic forms in L20, −2 , hence elements of Hm (N ), lie in L2k, −2 for all k ∈ N and thus are smooth.

33

The moduli space of deformations for a compact coassociative 4-fold P , by [23, Theorem 4.5], is smooth of dimension b2+ (P ). This suggests that we need to define an analogue of b2+ for an AC coassociative 4-fold N . Definition 7.3 As noted in Definition 7.2, H2 (N ) consists of smooth forms. 2 The Hodge star maps H2 (N ) into itself, so there is a splitting H2 (N ) = H+ (N )⊕ 2 H− (N ) where 2 H± (N ) = H2 (N ) ∩ C ∞ (Λ2± T ∗ N ).  2 (N ) , with φ2 as in Proposition 7.1. If [α], [β] ∈ Define J (N ) = φ2 Hcs J (N ), there exist compactly supported closed 2-forms ξ and η such that [α] = φ2 ([ξ]) and [β] = φ2 ([η]). We define a product on J (N ) × J (N ) by Z ξ ∧ η. (49) [α] ∪ [β] = N

Suppose that ξ ′ and η ′ are also compactly supported with [α] = φ2 ([ξ ′ ]) and [β] = φ2 ([η ′ ]). Then there exist 1-forms χ and ζ such that ξ − ξ ′ = dχ and η − η ′ = dζ. Therefore, Z Z Z ξ ∧ η − dχ ∧ η − ξ ′ ∧ dζ (ξ − dχ) ∧ (η − dζ) = ξ ′ ∧ η′ = N

N

N

=

Z

N



ξ ∧ η − d(χ ∧ η) − d(ξ ∧ ζ) =

Z

N

ξ ∧ η,

as both χ∧η and ξ ′ ∧ζ have compact support. The product (49) on J (N )×J (N ) is thus well-defined and is a symmetric topological product with a signature (a, b). By [15, Example 0.15], H2 (N ) ∼ = J (N ) via α 7→ [α]. We therefore define 2 b2+ (N ) = dim H+ (N ).

Hence, b2+ (N ) = a, which is a topological number. We can now prove a useful fact about the kernel of (d + d∗ )µ . Lemma 7.4 In the notation of Definitions 5.1 and 7.3, dim K(µ) ≤ b2+ (N ) ≤ dim K(−2) for all µ < −2 such that µ ∈ / D. Proof: By Theorem 3.5(a), recalling that p > 4 and k ≥ 2, Lpk+1, µ ֒→ L20, −2 and L2k+3, −2 ֒→ Lpk+1, −2 . The result follows from observations in Definitions 6.1 and 7.3.  We want to make a similar statement about the cokernel of (d + d∗ )µ . In fact, we can do a lot better, but we first need a result concerning functions on cones [5, Lemma 2.3]. 34

Proposition 7.5 Recall the cone C and its link Σ. Suppose that f : C → R is a nonzero function such that f (r, σ) = rµ fΣ (σ) for some function fΣ : Σ → R and µ ∈ R. Denoting the Laplacians on C and Σ by ∆C and ∆Σ respectively,  ∆C f (r, σ) = rµ−2 ∆Σ fΣ (σ) − µ(µ + 2)fΣ (σ) .

Therefore, since Σ is compact, µ(µ + 2) ≥ 0 and so there exist no nonzero homogeneous harmonic functions of order O(rµ ) on C with µ ∈ (−2, 0). Proposition 7.6 In the notation of Definition 5.9 and Theorem 5.10, dim C+ (µ) = dim C(µ) = b3 (N ) for all µ ∈ (−2, 0) \ D, and so they are independent of µ in this range. Proof: Let µ ∈ (−2, 0) \ D and let γ ∈ C+ (µ) ⊆ Lql+1, −3−µ (Λ3 T ∗ N ). Recall the closed compactly supported 1-form γˆ = ∗γ − dζ related to γ as given by Proposition 5.14. Clearly ∆ζ = −d∗ γˆ since d∗ ∗γ = 0. Hence, by Proposition 5.14, d∗ γˆ lies in the image of the map ∆−2−µ = ∆ : Lql+2, −2−µ (Λ0 T ∗ N ) → Lql, −4−µ (Λ0 T ∗ N ). Therefore d∗ γˆ is orthogonal, via the pairing given in Definition 3.2, to the kernel of the adjoint of ∆−2−µ . Let ν ∈ (−2, 0)\D. Proposition 7.5 implies, as shown in [5, Definition 2.5 & Theorem 2.11], that there are no elements of D(∆), defined in Theorem 5.3, between −2−µ and −2−ν. Thus Coker ∆−2−µ = Coker ∆−2−ν . Obviously, d∗ γˆ is then orthogonal to the kernel of the adjoint of ∆−2−ν . Consequently, there exists ζν ∈ Lql+2, −2−ν (Λ0 T ∗ N ) such that ∆−2−ν ζν = −d∗ γˆ and hence ζ − ζν is harmonic. Moreover, ζ − ζµ = o(ρ− min(µ , ν)−2 ) as ρ → ∞, where ρ is a radius function on N . Since − min(µ, ν) − 2 < 0, we use the Maximum Principle to deduce that ζ − ζµ = 0. Hence ∗γ and γ lie in Lql+1, −ν−3 for any ν ∈ (−2, 0). The dimension of C+ (µ) is therefore constant for µ ∈ (−2, 0) \ D. 3 We now note that, by [15, Example 0.15], H3 (N ) ∼ (N ) via η 7→ [η], = HdR q where H3 (N ) is given by (48). By Theorem 3.5(a), Ll+1, −3−µ ֒→ L20, −2 = L2 for µ ≥ −1 and L2l+1, −2 ֒→ Lql+1, −3−µ if µ < −1. By Theorem 5.10, C+ (µ) = C(µ). Hence, C+ (µ) ⊆ H3 (N ) whenever µ ∈ [−1, 0) \ D, and H3 (N ) ⊆ C+ (µ) if µ ∈ (−2, −1) \ D. The result follows. 

35

7.2

Rates λ ∈ [−2, 0) \ D

For convenience we introduce some extra notation. Definition 7.7 Since D is discrete by Proposition 5.4, we may choose λ− < −2 and λ+ ∈ (−2, 0) such that [λ− , λ+ ] ∩ D ⊆ {−2}. We are thus able to make the following proposition, which determines the dimension of M(N, λ) in a special case. Proposition 7.8 Use the notation from Definitions 5.5 and 7.3. If b1 (Σ) = 0 and λ ∈ [−2, 0) \ D, X d(µ). dim M(N, λ) = b2+ (N ) + µ∈D ∩(−2, λ)

Proof: Use the notation of Definition 7.7 and recall that −2 ∈ D if and only if b1 (Σ) > 0 by Proposition 5.4. Since −2 ∈ / D, [λ− , λ+ ] ∩ D = ∅. By Proposition 5.7, the dimensions of the kernel and cokernel of (19) are constant for µ ∈ [λ− , λ+ ]. Therefore, by Proposition 7.4, dim K(µ) = dim K(−2) = b2+ (N )

for µ ∈ [λ− , λ+ ].

Applying Propositions 5.6 and 7.6 completes the proof.



The next proposition enables us to calculate the dimension of M(N, λ) when −2 ∈ D. The proof is long, but the main idea is to identify the forms which are added to the kernel of (19) as the rate crosses −2. By the work in [16], these forms are precisely those which are asymptotic to homogeneous solutions ξ ∈ C ∞ (Λ2+ T ∗ C ⊕ Λ4 T ∗ C) of order O(r−2 ) to (d + d∗ )ξ = 0, as described after Proposition 5.4. In the course of the proof, we will need an elementary lemma. Lemma 7.9 If P is a compact 4-dimensional Riemannian manifold,   d C ∞ (Λ2 T ∗ P ) = d C ∞ (Λ2+ T ∗ P ) .

Proof: Let θ ∈ C ∞ (Λ2 T ∗ P ). By Hodge theory, there exists ψ ∈ C ∞ (Λ1 T ∗ P ) and ω ∈ C ∞ (Λ2 T ∗ P ), with dω = 0, such that θ = ∗dψ + ω. (Here, ω is the sum of an exact form and a harmonic form.) Therefore, dθ = d∗ dψ. Letting  Θ = ∗dψ + dψ ∈ C ∞ (Λ2+ T ∗ P ), we have that dθ = dΘ. Proposition 7.10 Use the notation of Definitions 7.3 and 7.7. If µ ∈ [λ− , −2), dim K(µ) = dim K(−2) = b2+ (N ). If ν ∈ (−2, λ+ ], dim K(ν) = b2+ (N ) − b0 (Σ) + b1 (Σ) + b0 (N ) − b1 (N ) + b3 (N ). 36

Proof: Let µ ∈ [λ− , −2) and ν ∈ (−2, λ+ ]. By Theorem 3.5(a), Lpk+1, µ ֒→ Lpk+1, −2 ֒→ Lpk+1, ν , so K(µ) ⊆ K(−2) ⊆ K(ν). Recall the map (d + d∗ )∞ defined by (22) and the bundle VΣ given in (23). By Definition 5.5, d(−2) is the dimension of the vector space of solutions ξ(t, σ) to (d + d∗ )∞ ξ = 0 of the form ξ(t, σ) = e−2t p(t, σ), where (t, σ) ∈ (T, ∞) × Σ and p(t, σ) is a polynomial in t taking values in C ∞ (VΣ ⊗ C). Furthermore, by Proposition 5.8 and its proof, d(−2) = b1 (Σ) and ξ must be of the form  ξ(t, σ) = e−2t α(σ) + dt ∧ ∗Σ α(σ) ,

where α is a closed and coclosed 2-form on Σ and ∗Σ is the Hodge star on Σ. Suppose α 6= 0. We shall see that the forms ξ of this type correspond to forms on N which add to the kernel of (19) as the rate crosses −2. First, however, we must transform from the cylindrical picture to the conical one. Remember, in (21), we related (d + d∗ )∞ to the usual operators d and d∗ acting on m-forms on N \ K ∼ = (R, ∞) × Σ. From this equation, since ξ is a 2-form, we notice that we need to consider η = e2t ξ as the form on the cone related to ξ. Changing coordinates (t, σ) to conical coordinates (r, σ), where r = et , we may write η(r, σ) = α(σ) + r−1 dr ∧ ∗Σ α(σ).

(50)

We must now consider possible extensions of η to a form on N . Recall that Ψ : (R, ∞) × Σ ∼ = N \ K and that ρ is a radius function on N as in Definition 2.7. We also remind the reader of Proposition 4.4: that there is an isomorphism  ˜ : Λ2+ T ∗ (R, ∞) × Σ → Λ2+ T ∗ (N \ K). Υ

By the work in [16, §5], as [λ− , λ+ ] ∩ D = {−2}, γ ∈ K(ν) if and only if there ˜ exist η of the form (50), and ζ ∈ K(ν) with ζ asymptotic to Υ(η), such that γ − ζ ∈ K(µ). We can calculate: |η(r, σ)|gcone = O(r−2 )

as r → ∞,

where gcone is the conical metric on (R, ∞) × Σ as given in Definition 2.5, since ˜ α is a 2-form independent of r. Thus, a form ζ asymptotic to Υ(η) must satisfy −2 ζ = O(ρ ) as ρ → ∞, so ζ ∈ / K(−2). Hence, γ ∈ / K(−2). 37

Use the notation of Proposition 7.1. We show that a form ζ as in the previous  2 paragraph will exist if and only if [α] ∈ p2 HdR (N ) . Necessity is clear, so we  2 (N ) with α the unique harmonic only consider sufficiency. Let [α] ∈ p2 HdR representative in the class. Note that the map πΣ : (R, ∞) × Σ ∼ = N \K → Σ 2 2 induces an isomorphism q2 : HdR (Σ) → HdR (N \ K). Now, η, given by (50), is self-dual with respect to gcone , so  ˜ ζ ′ = Υ(η) ∈ C ∞ Λ2+ T ∗ (N \ K) .

Moreover, since

η = α + d(∗Σ α log r),   2 2 we see that [ζ ′ ] = [η] = [α] in HdR (R, ∞) × Σ . Thus, q2−1 ([ζ ′ ]) ∈ p2 HdR (N ) . By the long exact sequence (47), the image of p2 is equal to the kernel of ∂2 .  Hence, if f ∈ C ∞ (N ) such that f = 0 on K and f = 1 on Ψ (R + 1, ∞) × Σ , 3 [d(f ζ ′ )] = 0 in Hcs (N ). Therefore, there exists a smooth, compactly supported, self-dual 2-form χ on N such that d(f ζ ′ ) = dχ by Lemma 7.9. We define ζ = f ζ ′ − χ, which is a closed self-dual 2-form on N asymptotic ˜ ˜ to Υ(η), since it equals Υ(η) outside the support of 1 − f and the compactly supported form χ. We have therefore shown that the space of kernel forms added at rate −2 is isomorphic to the image of p2 . We can calculate the dimension of this image by (47):  2 dim p2 HdR (N ) = −b0 (Σ) + b1 (Σ) + b0 (N ) − b1 (N ) + b3 (N ).

By Proposition 5.7, there are no changes in K(µ) for µ ∈ [λ− , −2) and the argument thus far shows that the kernel forms added at rate −2 do not lie in K(−2). We conclude that K(µ) = K(−2) for µ ∈ [λ− , −2), so dim K(µ) = dim K(−2) = b2+ (N ) by Lemma 7.4. The latter part of the proposition follows from the formula and arguments above, since K(ν) does not alter for ν ∈ (−2, λ+ ] by Proposition 5.7.  Note By (47), −b0 (Σ) + b1 (Σ) + b0 (N ) − b1 (N ) + b3 (N ) = 0 if b1 (Σ) = 0.

Proposition 7.10 shows that the function k(µ) = dim K(µ) is lower semicontinuous at −2 and is continuous there if −2 ∈ / D by the note above. Our next result shows that c(µ) = dim C+ (µ) is upper semi-continuous at −2. Proposition 7.11 Use the notation of Definition 7.7. If ν ∈ (−2, λ+ ], then dim C+ (ν) = dim C+ (−2) = b3 (N ). If µ ∈ [λ− , −2), dim C+ (µ) = b0 (Σ) − b0 (N ) + b1 (N ). 38

Proof: We know the index of (19) at rate ν by Propositions 7.6 and 7.10: iν (d + d∗ ) = b2+ (N ) − b0 (Σ) + b1 (Σ) + b0 (N ) − b1 (N ). By Proposition 5.6, recalling that d(−2) = b1 (Σ), iµ (d + d∗ ) = b2+ (N ) − b0 (Σ) + b0 (N ) − b1 (N ). Using Proposition 7.5 again we get the dimension of C+ (µ) claimed. The only part left to do is to show that dim C+ (−2) = b3 (N ). Remember that if the rate for the kernel forms is κ then the ‘dual’ rate for the cokernel forms is −4 − (κ − 1) = −3 − κ. Thus, by the work in [16], the forms γ which are added to the cokernel as the rate crosses −2 from above must be asymptotic to homogeneous forms on the cone of order O(r−1 ). Therefore, γ∈ / C(−2) ⊆ Lql+1, −1 (Λ3 T ∗ N ) as required.  Note The difference dim C+ (µ) − dim C+ (ν), as given in Proposition 7.11, is  1 equal to dim p1 HdR (N ) as defined in Proposition 7.1. We can prove an isomorphism between the image of p1 and the cokernel forms added at −2 in a similar manner to the proof of Proposition 7.10, but we omit it. Proposition 5.6, along with Propositions 7.6, 7.10 and 7.11, give us the dimension of K(λ), hence M(N, λ), for all λ ∈ [−2, 0) \ D. This is because the dimension of the cokernel is constant for these choices of λ, so the change of index of (19) equals the change in the dimension of K(λ). Proposition 7.12 In the notation of Definitions 5.5 and 7.3, if λ ∈ [−2, 0)\D, X d(µ). dim M(N, λ) = b2+ (N ) − b0 (Σ) + b1 (Σ) + b0 (N ) − b1 (N ) + b3 (N ) + µ∈D ∩(−2, λ)

7.3

Rates λ ∈ [0, 1) \ D

We now discuss the case λ ≥ 0 and begin by studying the point 0 ∈ D. Recall that d(0), as given by Proposition 5.5, is equal to the dimension of D(0) = {(α, β) ∈ C ∞ (Λ2 T ∗ Σ ⊕ Λ3 T ∗ Σ) : dΣ α = 0 and dΣ ∗Σ α + d∗Σ β = 2α}, using the notation of Proposition 5.4. It is clear, using integration by parts, that the equations which (α, β) ∈ D(0) satisfy are equivalent to dΣ ∗Σ α = 2α and d∗Σ β = 0. 39

The latter equation corresponds to constant 3-forms on Σ, so the solution set has dimension equal to b0 (Σ). If we define Z = {α ∈ C ∞ (Λ2 T ∗ Σ) : dΣ ∗Σ α = 2α},

(51)

then d(0) = b0 (Σ) + dim Z. Suppose that β ∈ C ∞ (Λ3 T ∗ Σ) satisfies d∗Σ β = 0 and corresponds to a form on N which adds to the kernel of (19) at rate 0. There exists, by [16, §5], a 4-form ζ on N asymptotic to r3 dr ∧ β on N \ K ∼ = (R, ∞) × Σ, and a self-dual 2-form ξ of order o(1) as ρ → ∞ such that dξ + d∗ ζ = 0, where ρ is a radius function on N . Since d∗ ζ is exact, ∗ζ is a harmonic function which is asymptotic to a function c, constant on each end of N , as given in Definition 2.6. Applying [5, Theorem 7.10] gives a unique harmonic function f on N which converges to c with order O(ρµ ) for all µ ∈ (−2, 0). The theorem cited is stated for an AC special Lagrangian submanifold L, but only uses the fact that L is an AC manifold and hence is applicable here. Therefore, ∗ζ − f = o(1) as ρ → ∞ and hence, by the Maximum Principle, ∗ζ = f . We deduce that d∗ ζ and dξ are O(ρ−3+ǫ ) as ρ → ∞ for any ǫ > 0 small, hence they lie in L2 . Integration by parts, now justified, shows that d∗ ζ = 0 and we conclude that ∗ζ is constant on each component of N . Hence, the piece of b0 (Σ) in d(0) that adds to the dimension of the kernel is equal to b0 (N ). The other 3-forms β on Σ satisfying d∗Σ β = 0 must correspond to cokernel forms, so b0 (Σ) − b0 (N ) is subtracted from the dimension of the cokernel at 0. For each end of N , we can define self-dual 2-forms of order O(1) given by ∂ translations of it, written as ∂x · ϕ for j = 1, . . . , 7. If the end is a flat R4 , we j only get three such self-dual 2-forms from it. So, if k ′ is the number of ends which are not 4-planes, dim Z ≥ 7k ′ + 3(b0 (Σ) − k ′ ) = 3b0 (Σ) + 4k ′ . Moreover, the translations of the components of N must correspond to kernel forms, since they are genuine deformations of N . If a component of N is a 4-plane then there are only three nontrivial translations of it. Therefore, if k is the number of components of N which are not 4-planes, at least 3b0 (N ) + 4k is added to the dimension of the kernel at rate 0 from dim Z. From this discussion, we can now state and prove the following inequalities for the dimension of M(N, λ) for λ ∈ [0, 1) \ D. 40

Proposition 7.13 Use the notation from Proposition 5.5, Definition 7.3 and Theorem 6.4. If λ ∈ [0, 1) \ D, X d(µ) dim M(N, λ) ≤ dim K(0) + b0 (N ) + dim Z − dim B(λ) + µ∈D ∩(0, λ)

where Z is defined in (51) and X

dim K(0) = b2+ (N ) − b0 (Σ) + b1 (Σ) + b0 (N ) − b1 (N ) + b3 (N ) +

d(µ).

µ∈D ∩(−2, 0)

Moreover, if k is the number of components of N which are not a flat R4 , X d(µ). dim M(N, λ) ≥ dim K(0) + b0 (Σ) + 3b0 (N ) + 4k − b3 (N ) − dim B(λ) + µ∈D ∩(0, λ)

Proof: By Theorem 6.4, dim M(N, λ) = dim K(λ) − dim B(λ). Using Propositions 5.6, 7.6 and 7.10 and the notation of Definition 7.7, we calculate: X d(µ) iλ (d + d∗ ) = iλ+ (d + d∗ ) + µ∈D ∩(−2, λ)

= b2+ (N ) − b0 (Σ) + b1 (Σ) + b0 (N ) − b1 (N ) +

X

d(µ)

µ∈D ∩(−2, λ)

= dim K(0) − b3 (N ) + d(0) +

X

d(µ).

µ∈D ∩(0, λ)

(An argument using asymptotic behaviour of kernel forms as in the proof of Proposition 7.10 leads to lower semi-continuity of k(µ) = dim K(µ) at 0). From the discussion above, the part of d(0) = b0 (Σ) + dim Z that subtracts from the cokernel of (19) as the rate crosses 0 is greater than or equal to b0 (Σ) − b0 (N ). Therefore, dim C+ (λ) ≤ b3 (N ) − b0 (Σ) + b0 (N ) using Proposition 7.6, so dim K(λ) = iλ (d + d∗ ) + dim C+ (λ)

≤ iλ (d + d∗ ) + b3 (N ) − b0 (Σ) + b0 (N ) X d(µ). = dim K(0) + b0 (N ) + dim Z + µ∈D ∩(0, λ)

The first inequality follows. Now, from the discussion above, at least b0 (N ) + 3b0 (N ) + 4k is added to the kernel of (19) from d(0) as the rate crosses 0. Remember dim C+ (λ) ≤ b3 (N ) − b0 (Σ) + b0 (N ). Therefore, at most this amount could be added to the

41

index of (19) from the sum of d(µ) terms without adding to the kernel. Thus, X d(µ) dim K(λ) ≥ dim K(0) + 4b0 (N ) + 4k − b3 (N ) + µ∈D ∩(0, λ)

− b3 (N ) + b0 (Σ) − b0 (N )

= dim K(0) + b0 (Σ) + 3b0 (N ) + 4k − b3 (N ) +

X

d(µ),

µ∈D ∩(0, λ)

from which we can easily deduce the second inequality.

7.4



Rates λ < −2, λ ∈ /D

By Theorem 6.4 and its proof, we can think of the map π as a projection from an open neighbourhood of 0 in the infinitesimal deformation space I(N, λ) to the obstruction space O(N, λ). Thus, the expected dimension of the moduli space is dim I(N, λ) − dim O(N, λ). We now find a lower bound for this dimension. Proposition 7.14 Use the notation of Definitions 6.1, 6.2 and 7.3. The expected dimension of the moduli space satisfies dim I(N, λ) − dim O(N, λ) ≥

b2+ (N ) − b0 (Σ) + b0 (N ) − b1 (N ) + b3 (N ) −

X

d(µ).

µ∈D ∩(λ, −2)

Proof: In Definition 6.2 we noted that dim O(N, λ) = dim C+ (λ) − dim C(λ), where C(λ) is given in (33). Since Lql+1, −1 ֒→ Lql+1, −3−λ by Theorem 3.5(a) (as λ > −2), we deduce that dim C(λ) ≥ b3 (N ) by Proposition 7.6. Therefore, dim I(N, λ) − dim O(N, λ) ≥ dim K(λ) − dim C+ (λ) + b3 (N ). Applying Propositions 5.6, 7.10 and 7.11 completes the proof.



Remark Any nonplanar coassociative 4-fold which is AC with rate λ has a 1-dimensional family of nontrivial AC coassociative deformations given by dilations. The obvious question is: where does this appear in the calculation of the dimension of the moduli space? The author believes that it should come from kernel forms of rate µ, where µ is the greatest element of D ∩ (−∞, λ).

42

8

Invariants

Suppose for this section that N is a coassociative 4-fold which is AC to a cone C in R7 with rate λ < −2 and suppose, for convenience, that N is connected. 2 Recall the space H+ (N ) and topological invariant b2+ (N ) given in Definition 7.3. Consider the deformation N 7→ et N for t ∈ R. Clearly et N is coassociative and AC to C with rate λ for all t ∈ R. Let u be the dilation vector field on R7 as given in (44). Define a self-dual 2-form αu on N , as in the note after Proposition 2.3, by αu = (u · ϕ)|T N . (52) The deformation corresponding to tαu , as given in Definition 4.6, is et N . We calculate, using (45), dαu = d(u · ϕ)|T N = 3ϕ|T N = 0 2 since N is coassociative. As λ < −2, αu ∈ L2 (Λ2+ T ∗ N ) and hence tαu ∈ H+ (N ) 2 t for all t ∈ R. So, there is a 1-dimensional subspace of H+ (N ) whenever e N 6= N for some t 6= 0; that is, when N is not a cone. Consequently, b2+ (N ) = 0 forces N∼ = R4 , as N is nonsingular. This discussion leads to our first invariant.

Definition 8.1 Let N be an AC coassociative 4-fold in R7 with rate λ < −2, define αu by (52) and let X(N ) = kαu k2L2 . By the discussion above, this is a well-defined invariant of N . We now consider, for Γ a 2-cycle in N and D a 3-cycle in R7 such that R ∂D = Γ, the integral D ϕ. Suppose D′ is another 3-cycle with ∂D′ = Γ. Then ∂(D − D′ ) = 0, so Z ϕ = [ϕ] · [D − D′ ] = 0 D−D′

R R 3 since [ϕ] ∈ HdR (R7 ) is zero. Therefore D ϕ = D′ ϕ; that is, the integral is independent of the choice of 3-cycle with boundary Γ. Suppose instead that Γ′ is a 2-cycle in N such that Γ − Γ′ = ∂E for some 3-cycle E ⊆ N . Let D and D′ be 3-cycles such that ∂D = Γ and ∂D′ = Γ′ . Then ∂D = ∂(E + D′ ) = Γ and thus Z Z Z Z Z ϕ, ϕ= ϕ+ ϕ= ϕ= D

E+D′

E

D′

D′

since E ⊆ N and ϕ|N = 0. Hence, we have shown the integral only depends on the homology class [Γ]. 43

We can now define our second invariant for any coassociative 4-fold in R7 . Definition 8.2 Let N be a coassociative 4-fold in R7 . Let Γ be a 2-cycle in N 2 and let D be a 3-cycle in R7 such that ∂D = Γ. Define [Y (N )] ∈ HdR (N ) by: Z [Y (N )] · [Γ] = ϕ. D

The work above shows that this is well-defined. In the notation of Definition 8.2, we calculate, using (45) and (52), Z Z Z Z αu = [αu ] · [Γ]. u·ϕ = d(u · ϕ) = ϕ= 3 D

Γ

Γ

D

Hence, 3[Y (N )] = [αu ]. We remind the reader of Definitions 7.2 and 7.3, in particular that J (N ) is 2 2 2 the image of Hcs (N ) in HdR (N ). We showed that αu ∈ H+ (N ) and we know 2 ∼ that H (N ) = J (N ) via γ 7→ [γ]. Thus, [Y (N )] and [αu ] lie in J (N ). Using the product (49), we deduce that Z αu ∧ αu = kαu k2L2 = X(N ). 9[Y (N )]2 = [αu ] ∪ [αu ] = N

We have used the fact that if αu = η + dξ, for some compactly supported closed ∞ 2-form η and ξ ∈ Cλ+1 (Λ1 T ∗ N ), then an integration by parts argument, valid since λ < −2, shows that Z Z αu ∧ αu = η ∧ η. N

N

We may thus derive a test which determines whether N is a cone. Proposition 8.3 Let N be a connected coassociative 4-fold in R7 which is AC with rate λ < −2. Use the notation of Definitions 7.3, 8.1 and 8.2. If b2+ (N ) = 0, N is a cone and hence a linear R4 in R7 as N is nonsingular. Moreover, N is a cone if and only if X(N ) = 0 or, equivalently, [Y (N )]2 = 0. We note, for interest, that a similar argument holds for a connected special Lagrangian m-fold L in Cm (m ≥ 2) which is AC to a cone C ∼ = (0, ∞) × Σ with rate λ < −m/2, in the following sense. Let u be the dilation vector field on Cm , and let ωm and Ωm be the symplectic and holomorphic volume forms on Cm respectively. Define βu = (u · ωm )|T L

and γu = (u · Ωm )|T L , 44

which are both zero if and only if L is a cone. By the proof of [23, Theorem 3.6], βu and γu are closed and γu = ∗βu . Moreover, by (3) and Theorem 3.5(a), βu and ∗βu lie in L2 as λ < −m/2. By [15, Example 0.15], the closed and coclosed (m − 1)-forms in L2 uniquely m−1 represent the cohomology classes in HdR (L). Therefore, if bm−1 (L) = 0, L is m m a cone and hence a linear R in C as L is nonsingular. Moreover, we may define the invariant X(L) = kβu k2L2 in an analogous way to X(N ) and see that L is a cone if and only if X(L) = 0. m−1 1 We define two invariants, [Y (L)] ∈ HdR (L) and [Z(L)] ∈ HdR (L) by Z Z [Y (L)] · [Γ] = ωm and [Z(L)] · [Γ′ ] = Im Ωm , D

D′

where Γ is a 1-cycle in L, D is a 2-cycle in Cm such that ∂D = Γ, Γ′ is an (m − 1)-cycle in L and D′ is an m-cycle in Cm such that ∂D′ = Γ′ . Analogues of these invariants on Σ were introduced in [5, Definition 7.2]. Using similar calculations from our discussion of [Y (N )], we can show that 2[Y (L)] = [βu ] and m[Z(L)] = [∗βu ]. By [5, Proposition 7.3], [Y (L)] lies in the image of the compactly supported cohomology for rates less than −1. Therefore, for λ < −m/2 ≤ −1 (as m ≥ 2), we calculate: Z 2m[Y (L)] ∪ [Z(L)] = [βu ] ∪ [∗βu ] = βu ∧ ∗βu = kβu k2L2 = X(L), L

m 1 (L), (L) and HdR where the cup product is given by the pairing between Hcs since [βu ] can be represented by a compactly supported closed 1-form. Hence X(L) = 0 if and only if [Y (L)] ∪ [Z(L)] = 0. We write these observations as a proposition.

Proposition 8.4 Let L be a connected special Lagrangian m-fold in Cm (m ≥ 2) which is AC with rate λ < −m/2. If bm−1 (L) = 0, L is a cone and hence a linear Rm in Cm as L is nonsingular. Moreover, in the notation above, L is a cone if and only if X(L) = 0 or, equivalently, [Y (L)] ∪ [Z(L)] = 0.

9

Examples

Before discussing examples we prove a general result which shows that if an AC coassociative 4-fold converges sufficiently quickly to a cone with symmetries, then it inherits those symmetries. The analogous statement is already known for AC special Lagrangian submanifolds.

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Proposition 9.1 Let N be a coassociative 4-fold which is asymptotically conical to a cone C in R7 with rate λ < −2. Let G be the identity component of the automorphisms of C, Aut(C) ⊆ G2 . Then N is G-invariant. Proof: Suppose for a contradiction that N is not G-invariant. Then there exists a vector field v in the Lie algebra g of G which preserves C but not N . Since v does not preserve N , it defines a nonzero closed self-dual 2-form αv = (v · ϕ)|T N as in the note after Proposition 2.3. Morever, αv ∈ Cλ∞ (Λ2+ T ∗ N ) since the deformation of N defined by αv , as in Definition 4.6, converges to C at the same rate as N , recalling that v preserves C. Notice that, since v ∈ g, G ⊆ G2 and ϕ is invariant under G2 , the class of ϕ in H 3 (R7 , N ), which is zero, is unchanged under flow along v. Let u be the dilation vector field on R7 as given in (44). This leads us to define βv ∈ C ∞ (Λ1 T ∗ N ) by βv (x) =

1 ϕ(v, u, x) 3

for x ∈ T N . Thus, by (45), dβv = αv and, since αv ∈ Cλ∞ (Λ2+ T ∗ N ) and ∞ u = O(r) as r → ∞, where r is the radial distance in R7 , βv ∈ Cλ+1 (Λ1 T ∗ N ). ∗ Since αv = dβv is closed and self-dual, d dβv = 0. If ρ is a radius function on N as in Definition 2.7, dβv = O(ρλ ) and βv = O(ρλ+1 ) as ρ → ∞. As L20, −2 = L2 by (3) and λ < −2, we see from Theorem 3.5(a) that dβv ∈ L2 and the following integration by parts argument is valid: kαv k2L2 = kdβv k2L2 = hd∗ dβ, βiL2 = 0. Therefore, αv = 0, our required contradiction.

9.1



SU(2)-invariant AC coassociative 4-folds

The coassociative 4-folds we discuss were constructed in [3, Theorem IV.3.2]. The construction involves the consideration of R7 as the imaginary part of the octonions, O, and the definition of an action of SU(2) on Im O as unit elements in the quaternions, H. We first define the action then state the result. Definition 9.2 Write Im O ∼ = Im H ⊕ H. Define an action on Im O by (x, y) 7→ (qx¯ q , q¯y) for q ∈ H such that |q| = 1. This defines an action of SU(2) on Im O ∼ = R7 .

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Proposition 9.3 Use the notation of Definition 9.2. Let (e, f ) ∈ Im H ⊕ H ∼ = Im O be such that |e| = |f | = 1 and let c ∈ R. Then Mc = {(sqe¯ q , r¯ q f ) : q ∈ H, |q| = 1, and s(4s2 − 5r2 )2 = c for r ≥ 0, s ∈ R} is an SU(2)-invariant coassociative 4-fold in R7 ∼ = Im O, where the action is given in Definition 9.2. Moreover, every coassociative 4-fold in R7 invariant under this SU(2) action is of the form Mc for some choice of c, e and f . Suppose first that c = 0. So, s = 0, s =



5 2 r

or s = −

√ 5 2 r.

Let

M00 = {(0, r¯ q f ) : r ≥ 0} ∼ and = R4 ! ) ( √ 5 ± qe¯ q , q¯f : r > 0 ∼ M0 = r ± = (0, ∞) × S 3 , 2 so that M0 = M00 ⊔ M0+ ⊔ M0− . Thus, M0 is a SU(2)-invariant cone with three ends, two of which are diffeomorphic to cones on S 3 and one which is a flat R4 . Suppose now c 6= 0 and take c > 0 without loss of generality. This forces s > 0 and 4s2 − 5r2 6= 0. It is then clear that Mc has two components, Mc+ and Mc− , corresponding to 4s2 − 5r2 > 0 and 4s2 − 5r2 < 0 respectively.

The component Mc+ is AC to the cone √ M0+ . We calculate the rate as follows. 1 For large r, s is√ approximately equal to 25 r and hence 4s2 − 5r2 = O(r− 2 ). 3 Therefore s = 25 r + O(r− 2 ) and thus Mc+ converges with rate −3/2 to M0+ . For each r 6= 0 we have an S 3 orbit in Mc+ , but when r = 0 there is an S 2 orbit. 2 Therefore, topologically, Mc+ is an R2 bundle over S 2 . Hence HdR (Mc+ ) = R. Suppose, for a contradiction, that b2+ (Mc+ ) = 1, in the notation of Definition 7.3. Therefore, there exists a smooth, closed, self-dual 2-form in L2 = L20, −2 . The existence of this form, and the fact that the deformation theory of Mc+ is unobstructed by Corollary 6.5, means that there is a coassociative deformation ˜ c+ of Mc+ which is AC to M + with rate less than −2. However, M + is SU(2)M 0 0 ˜ + must itself be invariant under SU(2). invariant so, by Proposition 9.1, M c Proposition 9.3 describes the family of all coassociative 4-folds invariant under ˜ c+ must be AC with rate −3/2, our required contradiction. this action, so M Hence, b2+ (Mc+ ) = 0 and, since the SU(2) action has generic orbit S 3 , we conclude that Mc+ is isomorphic to the bundle O(−1) over CP1 . We now turn to Mc− . Here there are two ends, one of which has the same behaviour as the end of Mc+ and the other is where s → 0. For the latter case, we quickly see that s = O(r−4 ) and so Mc− converges at rate −4 to M00 . As the case r = 0 is excluded here, Mc− is topologically R × S 3 and converges with 2 rate −3/2 to M0+ and rate −4 to M00 at its two ends. Hence HdR (Mc− ) = 0. 47

Consequently, for c 6= 0, b2+ (Mc ) = 0. Notice also that the link Σ of the cone to which Mc converges has b1 (Σ) = 0. Applying Corollary 6.5 and Proposition 7.12, the moduli space M(Mc , −3/2) is a smooth manifold with dimension X d(µ), µ∈D∩(−2,−3/2)

using the notation of Proposition 5.4 and Definition 5.5. Note Fox [2, Example 9.2 & Theorem 9.3] showed that the examples given in Proposition 9.3 are special cases of a surface bundle construction over a pseudoholomorphic curve in S 6 with null-torsion (in Proposition 9.3 the curve is a totally geodesic 2-sphere). The examples he produces are also coassociative 4-folds N which are AC to some cone C with rate −3/2, but the link Σ of C may now have b1 (Σ) > 0 and b2+ (N ) may be non-zero.

9.2

2-ruled coassociative 4-folds

In [17], the author introduced the notion of 2-ruled 4-folds. A 4-dimensional submanifold M in Rn is 2-ruled if it admits a fibration over a 2-fold S such that each fibre is an affine 2-plane. As commented in [17, §3], if S is compact then M is AC with rate λ ≤ 0 to its asymptotic cone. Therefore, 2-ruled 4-folds can be examples of AC submanifolds which are not cones. Unfortunately, the examples in [17, §5] of 2-ruled 4-folds are not coassociative, but are calibrated submanifolds of R8 known as Cayley 4-folds. However, the author has derived a system of ordinary differential equations whose solutions define 2-ruled coassociative 4-folds invariant under a U(1) action. Though the solutions have not yet been found, it is clear that this system will provide AC examples. The result [4, Theorem 3.5.1] gives a method of construction for coassociative 4-folds in R7 which are necessarily 2-ruled. However, the examples in [4] of coassociative 4-folds are not AC in the notion of Definition 2.5 but rather in a much weaker sense. In [2], Fox studies 2-ruled coassociative 4-folds, with a particular focus on cones. His works leads to a number of techniques for constructing 2-ruled coassociative 4-folds. Recently the author ([20, Theorem 7.5]) has explicitly classified the 2-ruled coassociative cones and showed that the general cone can be constructed using a pseudoholomorphic curve in S 6 and holomorphic data on the curve. It is the hope of the author to use this result to produce nontrivial examples of AC 2-ruled coassociative 4-folds, using both the methods in [17] and other means.

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[14] S. Lang, Real Analysis, Addison-Wesley, Reading, Massachusetts, 1983. [15] R. B. Lockhart, Fredholm, Hodge and Liouville Theorems on Noncompact Manifolds, Trans. Amer. Math. Soc. 301, 1-35, 1987. [16] R. B. Lockhart and R. C. McOwen, Elliptic Differential Operators on Noncompact Manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. 12, 409-447, 1985. [17] J. Lotay, 2-Ruled Calibrated 4-folds in R7 and R8 , J. London Math. Soc. 74, 219-243, 2006. [18] J. D. Lotay, Coassociative 4-folds with Conical Singularities, Comm. Anal. Geom. 15, 891-946, 2008. [19] J. D. Lotay, Desingularization of Coassociative 4-folds with Conical Singularities, Geom. Funct. Anal. 18, 2055–2100, 2008. [20] J. D. Lotay, Ruled Lagrangian Submanifolds of the 6-Sphere, to appear in Trans. Amer. Math. Soc. [21] S. P. Marshall, Deformations of Special Lagrangian Submanifolds, DPhil thesis, Oxford University, Oxford, 2002. [22] V. G. Maz’ya and B. Plamenevskij, Elliptic Boundary Value Problems, Amer. Math. Soc. Transl. 123, 1–56, 1984. [23] R. C. McLean, Deformations of Calibrated Submanifolds, Comm. Anal. Geom. 6, 705-747, 1998. [24] T. Pacini, Deformations of Asymptotically Conical Special Lagrangian Submanifolds, Pacific J. Math. 215, 151-181, 2004. [25] S. Salur, Deformations of Asymptotically Cylindrical Coassociative Submanifolds with Moving Boundary, preprint, arXiv:math.DG/0601420. Jason D. Lotay University College High Street Oxford OX1 4BH [email protected]

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